Three Dimensional Geometry jee mains questions with solutions

Q 1 :

Each of the angles $β$ and $γ$ that a given line makes with the positive y-axes and z-axes, respectively, is half of the angle that this line makes with the positive x-axes. Then the sum of all possible values of the angle $β$ is [2025]

$π$
$\frac{π}{2}$
$\frac{3 π}{4}$
$\frac{3 π}{2}$

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(3)

Let the line makes angle $α$ with positive x-axis, then $β = \frac{α}{2}$ and $γ = \frac{α}{2}$

Now, $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

$\Rightarrow \cos^{2} α + 2 \cos^{2} \frac{α}{2} = 1$

$\Rightarrow \cos^{2} α + \cos α = 0$

$\Rightarrow \cos α (\cos α + 1) = 0$

$\Rightarrow \cos α = 0, - 1 \Rightarrow α = \frac{π}{2}, π$

Now, $β = \frac{α}{2} \Rightarrow β = \frac{π}{4}, \frac{π}{2}$

So, required sum $= \frac{π}{4} + \frac{π}{2} = \frac{3 π}{4}$ .

Q 2 :

Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, –1) and C = (2, 0, –2). Then among the statements

(S1) : $△$ ABC is an isosceles right angled triangle, and

(S2) : the area of $△$ ABC is $\frac{9 \sqrt{2}}{2}$ .

only (S1) is true
both are false
both are true
only (S2) is true

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(1)

Given, A(x, y, z) be a point in xy-plane. Let the point P(0, 3, 2), Q(2, 0, 3) and R(0, 0, 1)

The distance of the point AP = AQ = AR

$\Rightarrow {(x - 0)}^{2} + {(y - 3)}^{2} + {(z - 2)}^{2}$

$= {(x - 2)}^{2} + {(y - 0)}^{2} + {(z - 3)}^{2}$

$= {(x - 0)}^{2} + {(y - 0)}^{2} + {(z - 1)}^{2}$

In xy-plane, z = 0

So, $x^{2} - 4 x + y^{2} + 9 + 4 = x^{2} + y^{2} + 1 \Rightarrow - 4 x + 13 = 1 \Rightarrow x = 3$

And $x^{2} + y^{2} - 6 y + 9 + 4 = x^{2} + y^{2} + 1 \Rightarrow y = 2$

So, A(3, 2, 0), B(1, 4, –1) and C(2, 0, –2).

In $△$ ABC

$A B = \sqrt{1 + 4 + 4} = 3, B C = \sqrt{1 + 16 + 1} = \sqrt{18}, C A = \sqrt{1 + 4 + 4} = 3$

So, AB = AC and $A B^{2} + A C^{2} = {(B C)}^{2}$

$∴$ $△$ ABC is an isosceles right angled triangle.

So, (S1) is true.

Also, Area of $△ A B C = \frac{1}{2} \times (Base) (Height) = \frac{1}{2} \times 3 \times 3 = \frac{9}{2}$

So, (S2) is false.

Q 3 :

One vertex of a rectangular parallelepiped is at the origin O and the lengths of its edges along $x, y$ and $z$ axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to $z$ -axis, not passing through O or P is [2023]

$\frac{12}{5}$
$\frac{12}{5 \sqrt{5}}$
$\frac{12}{\sqrt{5}}$
$12 \sqrt{5}$

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(1)

$Equation of line O P, \frac{x - 0}{3 - 0} = \frac{y - 0}{4 - 0} = \frac{z - 0}{5 - 0}$
i.e., $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$

Now, equation of edge parallel to $z$ -axis passing through $(3, 0, 5)$ and having direction ratios $< 0, 0, 1 >$ is

$\frac{x - 3}{0} = \frac{y - 0}{0} = \frac{z - 5}{1}$

Here, $a_{1} = (0, 0, 0), a_{2} = (3, 0, 5); b_{1} = (3, 4, 5), b_{2} = (0, 0, 1)$

So, $\vec{a_{2}} - \vec{a_{1}} = 3 \hat{i} + 5 \hat{k}$

$\vec{b_{1}} \times \vec{b_{2}} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 4 & 5 \\ 0 & 0 & 1 \end{matrix} |$ $= \hat{i} (4 - 0) - \hat{j} (3 - 0) + \hat{k} (0 - 0) = 4 \hat{i} - 3 \hat{j}$

$∴$ $Required shortest distance =$ $| \frac{(3 \hat{i} + 5 \hat{k}) \cdot (4 \hat{i} - 3 \hat{j})}{| 4 \hat{i} - 3 \hat{j} |} |$

$= \frac{12}{\sqrt{16 + 9}} = \frac{12}{5} units .$

Q 4 :

The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(−2, −3, 5) and D(1, −6, −7) is equal to [2023]

$8 \sqrt{38}$
$48$
$54$
$9 \sqrt{38}$

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(1)

$The area of quadrilateral A B C D is equal to \frac{1}{2}$ $| \vec{A C} \times \vec{B D} |$ .

Now, $\vec{A C} = (- 2 \hat{i} - 3 \hat{j} + 5 \hat{k}) - (2 \hat{i} + \hat{j} + \hat{k}) = - 4 \hat{i} - 4 \hat{j} + 4 \hat{k}$

and $\vec{B D} = (\hat{i} - 6 \hat{j} - 7 \hat{k}) - (\hat{i} + 2 \hat{j} + 5 \hat{k}) = - 8 \hat{j} - 12 \hat{k}$

So, $\frac{1}{2}$ $| \vec{A C} \times \vec{B D} |$ $= \frac{1}{2}$ $| | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 4 & - 4 & 4 \\ 0 & - 8 & - 12 \end{matrix} | |$

$= \frac{1}{2}$ $| 80 \hat{i} - 48 \hat{j} + 32 \hat{k} |$ $= \frac{1}{2} \sqrt{9728} = 8 \sqrt{38}$

Q 5 :

The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to: [2023]

$2 \sqrt{3}$
$\sqrt{14}$
$3$
$\sqrt{6}$

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(2)

$Let Q = - 3 \hat{i} + 2 \hat{j} + 3 \hat{k}$

$\vec{P Q} = - 7 \hat{i} - 4 \hat{j} + 5 \hat{k}, \vec{b} = 3 \hat{i} + 3 \hat{j} - \hat{k} \Rightarrow$ $| \vec{b} |$ $= \sqrt{19}$

$\vec{P Q} \times \vec{b} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 7 & - 4 & 5 \\ 3 & 3 & - 1 \end{matrix} |$ $= - 11 \hat{i} + 8 \hat{j} - 9 \hat{k}$

$Required distance =$ $| \frac{\vec{P Q} \times \vec{b}}{| \vec{b} |} |$ $= \frac{\sqrt{11^{2} + 8^{2} + 9^{2}}}{\sqrt{19}} = \sqrt{14} units$

Q 6 :

Let the plane $x + 3 y - 2 z + 6 = 0$ meet the coordinate axes at the points A, B, C. If the orthocenter of the triangle ABC is $(α, β, \frac{6}{7})$ , then $98 {(α + β)}^{2}$ is equal to _____ . [2023]

(288)

Plane $x + 3 y - 2 z + 6 = 0$ meets the coordinate axes at the points A, B, C.

$For x-axis: x + 0 - 0 + 6 = 0 \Rightarrow x = - 6 i.e., (- 6, 0, 0)$

$For y-axis: 0 + 3 y - 0 + 6 = 0 \Rightarrow y = - 2 i.e., (0, - 2, 0)$

$For z-axis: 0 + 0 - 2 z + 6 = 0 \Rightarrow z = 3 i.e., (0, 0, 3)$

$\vec{A B} = 6 \hat{i} - 2 \hat{j}, \vec{B C} = 2 \hat{j} + 3 \hat{k}, \vec{A C} = 6 \hat{i} + 3 \hat{k}$

Now, $\vec{A P} \cdot \vec{B C} = 0$

$∴$ $(α + 6, β, \frac{6}{7}) \cdot (0, 2, 3) = 0$

$\Rightarrow 2 β + 3 \cdot \frac{6}{7} = 0 \Rightarrow β = \frac{- 9}{7}$

Similarly $\vec{C P} \cdot \vec{A B} = 0$

$(α, β, - \frac{15}{7}) \cdot (6, - 2, 0) = 0 \Rightarrow 6 α - 2 β = 0$

$\Rightarrow 6 α - 2 (\frac{- 9}{7}) = 0 \Rightarrow 6 α = \frac{- 18}{7} \Rightarrow α = \frac{- 3}{7}$

$∴$ $98 {(α + β)}^{2} = 98 {(\frac{- 3}{7} - \frac{9}{7})}^{2} = 98 {(\frac{- 12}{7})}^{2} = 98 \cdot \frac{144}{49} = 288$

Q 7 :

Let the direction cosines of two lines satisfy the equations $4 l + m - n = 0$ and $2 m n + 10 n l + 3 l m = 0 .$ Then the cosine of the acute angle between these lines is: [2026]

$\frac{10}{\sqrt{38}}$
$\frac{20}{3 \sqrt{38}}$
$\frac{10}{7 \sqrt{38}}$
$\frac{10}{3 \sqrt{38}}$

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(4)

$Direction cosines of two lines satisfy the equation$

$\Rightarrow 4 ℓ + m - n = 0 . . . (1)$

$2 m n + 10 n ℓ + 3 ℓ m = 0 . . . (2)$

$And we know$

$\Rightarrow ℓ^{2} + m^{2} - n^{2} = 1 . . . (3)$

$\Rightarrow n = 4 ℓ + m putting in eqn. (1)$

$\Rightarrow n (2 m + 10 ℓ) + 3 ℓ m = 0$

$\Rightarrow (4 ℓ + m) (2 m + 10 ℓ) + 3 ℓ m = 0$

$\Rightarrow 8 ℓ m + 40 ℓ^{2} + 2 m^{2} + 10 ℓ m + 3 ℓ m = 0$

$\Rightarrow 40 ℓ^{2} + 21 ℓ m + 2 m^{2} = 0$

$\Rightarrow (8 ℓ + m) (5 ℓ + 2 m) = 0$

$Case 1: 8 ℓ + m = 0 \Rightarrow m = - 8 ℓ$

$Case 2: 5 ℓ + 2 m = 0 \Rightarrow m = - \frac{5}{2} ℓ$

$So direction ratio of L_{1} is ℓ, - 8 ℓ, - 4 ℓ$

$and direction ratio of L_{2} is ℓ, \frac{- 5 ℓ}{2}, \frac{3 ℓ}{2}$

$\cos θ =$ $| \frac{ℓ^{2} + 20 ℓ^{2} - 6 ℓ^{2}}{\sqrt{ℓ^{2} + 64 ℓ^{2} + 16 ℓ^{2}} \sqrt{ℓ^{2} + \frac{25 ℓ^{2}}{4} + \frac{9 ℓ^{2}}{4}}} |$

$= \frac{15 ℓ^{2}}{(9 ℓ) \frac{\sqrt{38} ℓ}{2}}$ $= \frac{10}{3 \sqrt{38}}$

$Ans. = \frac{10}{3 \sqrt{38}}$

Q 8 :

Distance of the point $(α, β, γ)$ from Y-axis is :

$β$
$| β |$
$| β |$ $+$ $| γ |$
$\sqrt{α^{2} + γ^{2}}$

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(4)

Ans. $\sqrt{α^{2} + γ^{2}}$

Explanation:

Required distance $= \sqrt{{(α - 0)}^{2} + {(β - β)}^{2} + {(γ - 0)}^{2}}$

$= \sqrt{α^{2} + γ^{2}}$

Q 9 :

If the direction cosines of a line are $k$ , $k$ and $k$ , then :

$k > 0$
$0 < k < 1$
$k = 1$
$k = \frac{1}{\sqrt{3}}$ or $- \frac{1}{\sqrt{3}}$

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(4)

Ans. $k = \frac{1}{\sqrt{3}}$ or $- \frac{1}{\sqrt{3}}$

Explanation:

Since, direction cosines of a line are $k$ , $k$ and $k$ .

$∴$ $l = k, m = k$ and $n = k$

We know that,

$l^{2} + m^{2} + n^{2} = 1$

$\Rightarrow$ $k^{2} + k^{2} + k^{2} = 1$

$\Rightarrow$ $k^{2} = \frac{1}{3}$

$∴$ $k = \pm \frac{1}{\sqrt{3}}$

Q 10 :

Under what condition do $(\frac{1}{\sqrt{2}}, \frac{1}{2}, k)$ represent direction cosines of a line?

$k = \frac{1}{2}$
$k = \pm \frac{1}{2}$
$k = - \frac{1}{2}$
$k$ can take any value

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Ans. $k = \pm \frac{1}{2}$

Explanation:

For $(\frac{1}{\sqrt{2}}, \frac{1}{2}, k)$ to represent direction cosines, we should have

${(\frac{1}{\sqrt{2}})}^{2} + {(\frac{1}{2})}^{2} + k^{2} = 1$

or $\frac{1}{2} + \frac{1}{4} + k^{2} = 1$

$k^{2} = 1 - \frac{3}{4}$

$\Rightarrow$ $k = \pm \frac{1}{2}$

Q 11 :

Any three numbers which are proportional to the direction cosines of a line, are called:

direction angles
direction ratios
another set of direction cosines
none of the above

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Ans. direction ratios

Explanation:

Any three numbers which are proportional to the direction cosines of a line, are called the direction ratios of the line. If $l, m$ and $n$ are direction cosines and $a, b$ and $c$ are direction ratios of a line, then $a = λ l, b = λ m$ and $c = λ n$ for any non-zero $λ \in R$ .

Q 12 :

Direction ratios of two lines are $a, b, c$ and $\frac{1}{b c}$ , $\frac{1}{c a}$ , $\frac{1}{a b}$ . The lines are :

Mutually perpendicular
Parallel
Coincident
None of the above

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(2)

Ans. Parallel

Explanation:

As, $\frac{a}{(1 / b c)} = \frac{b}{(1 / c a)} = \frac{c}{(1 / a b)}$

Hence, lines are parallel.

Q 13 :

Direction ratios of the line represented by the equation $x = a y + b, z = c y + d$ are :

$(a, 1, c)$
$(a, b - d, c)$
$(a, 1, a)$
$(b, a c, d)$

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(1)

Ans. $(a, 1, c)$

Explanation:

Given,

$x = a y + b, z = c y + d$

$∴$ $y = \frac{x - b}{a}, y = \frac{z - d}{c}$

$∴$ $\frac{x - b}{a} = \frac{y}{1} = \frac{z - d}{c}$

Hence direction ratios are $(a, 1, c)$ .

Q 14 :

If $α, β, γ$ are the angles that a line makes with the positive direction of $X$ , $Y$ , $Z$ axes, respectively, then the direction cosines of the line are :

$\sin α, \sin β, \sin γ$
$\cos α, \cos β, \cos γ$
$\tan α, \tan β, \tan γ$
$\cos^{2} α, \cos^{2} β, \cos^{3} γ$

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Ans. $\cos α, \cos β, \cos γ$

Explanation:

Cosines of the angles $α, β, γ$ made by a line with positive direction of $X$ , $Y$ , $Z$ axes are $\cos α, \cos β, \cos γ$ .

Q 15 :

The distance of a point $P (a, b, c)$ from X-axis is :

$\sqrt{a^{2} + c^{2}}$
$\sqrt{a^{2} + b^{2}}$
$\sqrt{b^{2} + c^{2}}$
$b^{2} + c^{2}$

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Ans. $\sqrt{b^{2} + c^{2}}$

Explanation:

The required distance is the distance of $P (a, b, c)$ from $Q (a, 0, 0)$ , which is $\sqrt{b^{2} + c^{2}} .$

Q 16 :

A line makes equal angles with coordinate axis. Direction cosines of this line are :

$\pm (1, 1, 1)$
$\pm (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$
$\pm (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$
$\pm (\frac{1}{\sqrt{3}}, \frac{- 1}{\sqrt{3}}, \frac{- 1}{\sqrt{3}})$

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Ans. $\pm (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$

Explanation:

Let the line make angle $α$ with each of the axis.

Then, its direction cosines are $\cos α, \cos α, \cos α$ .

Since

$\cos^{2} α + \cos^{2} α + \cos^{2} α = 1$

$3 \cos^{2} α = 1$

$\cos^{2} α = \frac{1}{3}$

$∴$ $\cos α = \pm \frac{1}{\sqrt{3}}$

Q 17 :

The vector equation of the symmetrical form of equation of straight line $\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}$ is:

$\vec{r} = (3 \hat{i} + 7 \hat{j} + 2 \hat{k}) + μ (5 \hat{i} + 4 \hat{j} - 6 \hat{k})$
$\vec{r} = (5 \hat{i} + 4 \hat{j} - 6 \hat{k}) + μ (3 \hat{i} + 7 \hat{j} + 2 \hat{k})$
$\vec{r} = (5 \hat{i} - 4 \hat{j} - 6 \hat{k}) + μ (3 \hat{i} - 7 \hat{j} - 2 \hat{k})$
$\vec{r} = (5 \hat{i} - 4 \hat{j} + 6 \hat{k}) + μ (3 \hat{i} + 7 \hat{j} + 2 \hat{k})$

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(4)

Ans. $\vec{r} = (5 \hat{i} - 4 \hat{j} + 6 \hat{k}) + μ (3 \hat{i} + 7 \hat{j} + 2 \hat{k})$

Explanation:

$\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$ have vector form

$\vec{r} = (x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k}) + λ (a \hat{i} + b \hat{j} + c \hat{k})$

Required equation in vector form is

$\vec{r} = (5 \hat{i} - 4 \hat{j} + 6 \hat{k}) + μ (3 \hat{i} + 7 \hat{j} + 2 \hat{k})$

Q 18 :

The lines in a space which are neither intersecting nor parallel, are called:

concurrent lines
intersecting lines
skew lines
parallel lines

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Ans. skew lines

Explanation:

In a space, there are lines which are neither intersecting nor parallel. Infact, such pair of lines are non-coplanar and are called skew-lines.

Q 19 :

The two lines $x = a y + b, z = c y + d$ and $x = a' y + b', z = c' y + d'$ will be perpendicular, if and only if:

$a a' + c c' + 1 = 0$
$a a' + b b' + c c' + 1 = 0$
$a a' + b b' + c c' = 0$
$(a + a') (b + b') + (c + c') = 0$

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(1)

Ans. $a a' + c c' + 1 = 0$

Explanation:

$x = a y + b$

$\Rightarrow$ $y = \frac{x - b}{a}$

$z = c y + d$

$\Rightarrow$ $y = \frac{z - d}{c}$

$\frac{x - b}{a} = \frac{y}{1} = \frac{z - d}{c}$ $\dots (i)$

Similarly, $\frac{x - b'}{a'} = \frac{y}{1} = \frac{z - d'}{c'}$ $\dots (i i)$

Lines are perpendicular so the sum of corresponding product of their dr's be zero. So for perpendicularity of lines, $a a' + 1 + c c' = 0 .$

Q 20 :

A line makes angles of $45 °$ and $60 °$ with the positive axis of X and Y respectively. The angle made by the same line with the positive axis of Z, is:

30° or 60°
60° or 90°
90° or 120°
60° or 120°

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Ans. 60° or 120°

Explanation:

Given $α = 45 °, β = 60 °, γ = ?$

$∵$ $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

$∴$ $\cos^{2} γ = 1 - \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$

$\Rightarrow$ $γ = 60 ° or 120 °$

Q 21 :

If $α, β, γ$ be the angles which a line makes with the coordinate axis, then:

$\sin^{2} α + \cos^{2} β + \sin^{2} γ = 1$
$\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$
$\sin^{2} α + \sin^{2} β + \sin^{2} γ = 1$
$\cos^{2} α + \cos^{2} β + \sin^{2} γ = 1$

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Ans. $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

Explanation:

If $α, β, γ$ be the angles which a line makes with the coordinate axes, then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1 .$

Q 22 :

The equation of a line passing through the point $(- 3, 2, - 4)$ and equally inclined to the axis, are:

$x - 3 = y + 2 = z - 4$
$\frac{x + 3}{1} = \frac{y - 2}{1} = \frac{z + 4}{1}$
$\frac{x + 3}{1} = \frac{y - 2}{2} = \frac{z + 4}{3}$
None of the above

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Ans. $\frac{x + 3}{1} = \frac{y - 2}{1} = \frac{z + 4}{1}$

Explanation:

Required equation of lines is

$\frac{x + 3}{1} = \frac{y - 2}{1} = \frac{z + 4}{1}$

Q 23 :

The equation of straight line passing through the point $(a, b, c)$ and parallel to Z-axis is:

$\frac{x - a}{1} = \frac{y - b}{1} = \frac{z - c}{0}$
$\frac{x - a}{0} = \frac{y - b}{1} = \frac{z - c}{1}$
$\frac{x - a}{1} = \frac{y - b}{0} = \frac{z - c}{0}$
$\frac{x - a}{0} = \frac{y - b}{0} = \frac{z - c}{1}$

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Ans. $\frac{x - a}{0} = \frac{y - b}{0} = \frac{z - c}{1}$

Explanation:

The line through $(a, b, c)$ is

$\frac{x - a}{l} = \frac{y - b}{m} = \frac{z - c}{n}$ $\dots (i)$

Since the line is parallel to Z-axis, therefore, the direction cosines are $(0, 0, 1) .$

Hence, the line will be $\frac{x - a}{0} = \frac{y - b}{0} = \frac{z - c}{1}$

Q 24 :

The length of the perpendicular from point $(1, 2, 3)$ to the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ is:

5
6
7
8

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Ans. 7

Explanation:

Let a point on a given line be $A (3 λ + 6, 2 λ + 7, - 2 λ + 7) .$

Let the point $B$ be $(1, 2, 3)$ .

$∴$ Direction ratios of AB $= (3 λ + 6 - 1, 2 λ + 7 - 2, - 2 λ + 7 - 3)$

$= (3 λ + 5, 2 λ + 5, - 2 λ + 4)$

Given, direction ratios of the line are $3, 2, - 2$ .

$∵$ Lines are perpendicular.

$∴$ $3 (3 λ + 5) + 2 (2 λ + 5) - 2 (- 2 λ + 4) = 0$

$\Rightarrow$ $9 λ + 15 + 4 λ + 10 + 4 λ - 8 = 0$

$\Rightarrow$ $17 λ + 17 = 0$

$\Rightarrow$ $λ = - 1$

$∴$ $Point A is (3, 5, 9) .$

$∴$ The length of perpendicular $= \sqrt{{(3 - 1)}^{2} + {(5 - 2)}^{2} + {(9 - 3)}^{2}}$

$= \sqrt{4 + 9 + 36}$

$= \sqrt{49}$

$= 7$

Q 25 :

Equation of X-axis is:

$\frac{x}{1} = \frac{y}{1} = \frac{z}{1}$
$\frac{x}{0} = \frac{y}{1} = \frac{z}{1}$
$\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$
$\frac{x}{0} = \frac{y}{0} = \frac{z}{1}$

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Ans. $\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$

Explanation:

Since, on X-axis coordinates of $y$ and $z$ are zero.

$∴$ Equation of X-axis becomes

$\frac{x - 0}{1 - 0} = \frac{y - 0}{0 - 0} = \frac{z - 0}{0 - 0}$

$\Rightarrow$ $\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$

Q 26 :

The equations of X-axis in space are:

$x = 0, y = 0$
$x = 0, z = 0$
$x = 0$
$y = 0, z = 0$

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Ans. $y = 0, z = 0$

Explanation:

On X-axis the $y$ -coordinate and $z$ -coordinate are zero.

Q 27 :

The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{4 - z}{k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$ are mutually perpendicular if the value of $k$ is:

$- \frac{2}{3}$
$\frac{2}{3}$
$- 2$
$2$

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(1)

Ans. $- \frac{2}{3}$

Explanation:

First line is,

$\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}$

$∴$ Direction ratios of first line $= 1, 1, - k$

Second line is,

$\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$

$∴$ Direction ratios of second line $= k, 2, - 2$

We know, two lines are perpendicular to each other, if the dot product of their direction ratios is zero.

$i.e.,$ $1 \times k + 1 \times 2 + (- k) \times (- 2) = 0$

$\Rightarrow$ $k + 2 + 2 k = 0$

$\Rightarrow$ $3 k = - 2$

$\Rightarrow$ $k = - \frac{2}{3}$

Q 28 :

The shortest distance between the lines $x = y + 2 = 6 z - 6$ and $x + 1 = 2 y = - 12 z$ is:

$\frac{1}{2}$
$2$
$1$
$\frac{3}{2}$

1

2

3

4

(2)

Ans. 2

Explanation:

The lines are $\frac{x}{6} = \frac{y + 2}{6} = \frac{z - 1}{1}$

and $\frac{x + 1}{12} = \frac{y}{6} = \frac{z}{- 1}$

Here, ${\vec{a}}_{1} = - 2 \hat{j} + \hat{k}, {\vec{b}}_{1} = 6 \hat{i} + 6 \hat{j} + \hat{k},$

${\vec{a}}_{2} = - \hat{i},$

${\vec{b}}_{2} = 12 \hat{i} + 6 \hat{j} - \hat{k}$

${\vec{b}}_{1} \times {\vec{b}}_{2} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 6 & 1 \\ 12 & 6 & - 1 \end{matrix} |$

$= - 12 \hat{i} + 18 \hat{j} - 36 \hat{k}$

$Shortest distance =$ $\frac{| ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) |}{| {\vec{b}}_{1} \times {\vec{b}}_{2} |}$

$= \frac{| (- \hat{i} + 2 \hat{j} - \hat{k}) \cdot (- 12 \hat{i} + 18 \hat{j} - 36 \hat{k}) |}{\sqrt{{(- 12)}^{2} + {(18)}^{2} + {(- 36)}^{2}}}$

$= \frac{| 12 + 36 + 36 |}{\sqrt{1764}} = \frac{84}{42} = 2$

Q 29 :

The distance of the plane $\vec{r} \cdot (\frac{2}{7} \hat{i} + \frac{3}{7} \hat{j} - \frac{6}{7} \hat{k}) = 1$ from the origin is:

$1$
$7$
$\frac{1}{7}$
None of these

1

2

3

4

(1)

Ans. 1

Explanation:

The distance of the plane $\vec{r} \cdot (\frac{2}{7} \hat{i} + \frac{3}{7} \hat{j} - \frac{6}{7} \hat{k}) = 1$ from the origin is $1$ .

[Since, $\vec{r} \cdot \vec{n} = d$ is the form of above equation, where $d$ represents the distance of plane from the origin i.e., $d = 1$ ]

Q 30 :

The locus represented by $x y + y z = 0$ is:

a pair of perpendicular lines
a pair of parallel lines
a pair of parallel planes
a pair of perpendicular planes

1

2

3

4

(4)

Ans. a pair of perpendicular planes

Explanation:

We have, $x y + y z = 0$

$\Rightarrow$ $x y = - y z$

So, a pair of perpendicular planes.

Topic Question Set

Q 1 :

Each of the angles $β$ and $γ$ that a given line makes with the positive y-axes and z-axes, respectively, is half of the angle that this line makes with the positive x-axes. Then the sum of all possible values of the angle $β$ is [2025]

Q 4 :

The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(−2, −3, 5) and D(1, −6, −7) is equal to [2023]

Q 5 :

The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to: [2023]

Q 6 :

Let the plane $x + 3 y - 2 z + 6 = 0$ meet the coordinate axes at the points A, B, C. If the orthocenter of the triangle ABC is $(α, β, \frac{6}{7})$ , then $98 {(α + β)}^{2}$ is equal to _____ . [2023]

Q 7 :

Let the direction cosines of two lines satisfy the equations $4 l + m - n = 0$ and $2 m n + 10 n l + 3 l m = 0 .$ Then the cosine of the acute angle between these lines is: [2026]

Q 8 :

Distance of the point $(α, β, γ)$ from Y-axis is :

Q 9 :

If the direction cosines of a line are $k$ , $k$ and $k$ , then :

Q 10 :

Under what condition do $(\frac{1}{\sqrt{2}}, \frac{1}{2}, k)$ represent direction cosines of a line?

Q 11 :

Any three numbers which are proportional to the direction cosines of a line, are called:

Q 12 :

Direction ratios of two lines are $a, b, c$ and $\frac{1}{b c}$ , $\frac{1}{c a}$ , $\frac{1}{a b}$ . The lines are :

Q 13 :

Direction ratios of the line represented by the equation $x = a y + b, z = c y + d$ are :

Q 14 :

If $α, β, γ$ are the angles that a line makes with the positive direction of $X$ , $Y$ , $Z$ axes, respectively, then the direction cosines of the line are :

Q 15 :

The distance of a point $P (a, b, c)$ from X-axis is :

Q 16 :

A line makes equal angles with coordinate axis. Direction cosines of this line are :

Q 17 :

The vector equation of the symmetrical form of equation of straight line $\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}$ is:

Q 18 :

The lines in a space which are neither intersecting nor parallel, are called:

Q 19 :

The two lines $x = a y + b, z = c y + d$ and $x = a' y + b', z = c' y + d'$ will be perpendicular, if and only if:

Q 20 :

A line makes angles of $45 °$ and $60 °$ with the positive axis of X and Y respectively. The angle made by the same line with the positive axis of Z, is:

Q 21 :

If $α, β, γ$ be the angles which a line makes with the coordinate axis, then:

Q 22 :

The equation of a line passing through the point $(- 3, 2, - 4)$ and equally inclined to the axis, are:

Q 23 :

The equation of straight line passing through the point $(a, b, c)$ and parallel to Z-axis is:

Q 24 :

The length of the perpendicular from point $(1, 2, 3)$ to the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ is:

Q 25 :

Equation of X-axis is:

Q 26 :

The equations of X-axis in space are:

Q 27 :

The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{4 - z}{k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$ are mutually perpendicular if the value of $k$ is:

Q 28 :

The shortest distance between the lines $x = y + 2 = 6 z - 6$ and $x + 1 = 2 y = - 12 z$ is:

Q 29 :

The distance of the plane $\vec{r} \cdot (\frac{2}{7} \hat{i} + \frac{3}{7} \hat{j} - \frac{6}{7} \hat{k}) = 1$ from the origin is:

Q 30 :

The locus represented by $x y + y z = 0$ is:

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Topic Question Set

Q 1 : Each of the angles β and γ that a given line makes with the positive y-axes and z-axes, respectively, is half of the angle that this line makes with the positive x-axes. Then the sum of all possible values of the angle β is [2025]

Q 2 : Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, –1) and C = (2, 0, –2). Then among the statements (S1) : △ABC is an isosceles right angled triangle, and (S2) : the area of △ABC is 922.

Q 4 : The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(−2, −3, 5) and D(1, −6, −7) is equal to [2023]

Q 5 : The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to: [2023]

Q 6 : Let the plane x+3y-2z+6=0 meet the coordinate axes at the points A, B, C. If the orthocenter of the triangle ABC is (α,β,67), then 98(α+β)2 is equal to _____ . [2023]

Q 7 : Let the direction cosines of two lines satisfy the equations 4l+m-n=0 and 2mn+10nl+3lm=0. Then the cosine of the acute angle between these lines is: [2026]

Q 8 : Distance of the point (α,β,γ) from Y-axis is :

Q 9 : If the direction cosines of a line are k, k and k, then :

Q 10 : Under what condition do (12,12,k) represent direction cosines of a line?

Q 11 : Any three numbers which are proportional to the direction cosines of a line, are called:

Q 12 : Direction ratios of two lines are a, b, c and 1bc, 1ca, 1ab. The lines are :

Q 13 : Direction ratios of the line represented by the equation x=ay+b, z=cy+d are :

Q 14 : If α,β,γ are the angles that a line makes with the positive direction of X, Y, Z axes, respectively, then the direction cosines of the line are :

Q 15 : The distance of a point P(a,b,c) from X-axis is :

Q 16 : A line makes equal angles with coordinate axis. Direction cosines of this line are :

Q 17 : The vector equation of the symmetrical form of equation of straight line x-53=y+47=z-62 is:

Q 18 : The lines in a space which are neither intersecting nor parallel, are called:

Q 19 : The two lines x=ay+b, z=cy+d and x=a'y+b', z=c'y+d' will be perpendicular, if and only if:

Q 20 : A line makes angles of 45° and 60° with the positive axis of X and Y respectively. The angle made by the same line with the positive axis of Z, is:

Q 21 : If α,β,γ be the angles which a line makes with the coordinate axis, then:

Q 22 : The equation of a line passing through the point (-3, 2, -4) and equally inclined to the axis, are:

Q 23 : The equation of straight line passing through the point (a,b,c) and parallel to Z-axis is:

Q 24 : The length of the perpendicular from point (1,2,3) to the line x-63=y-72=z-7-2 is:

Q 25 : Equation of X-axis is:

Q 26 : The equations of X-axis in space are:

Q 27 : The lines x-21=y-31=4-zk and x-1k=y-42=z-5-2 are mutually perpendicular if the value of k is:

Q 28 : The shortest distance between the lines x=y+2=6z-6 and x+1=2y=-12z is:

Q 29 : The distance of the plane r→·(27i^+37j^-67k^)=1 from the origin is:

Q 30 : The locus represented by xy+yz=0 is:

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Q 1 :

Each of the angles $β$ and $γ$ that a given line makes with the positive y-axes and z-axes, respectively, is half of the angle that this line makes with the positive x-axes. Then the sum of all possible values of the angle $β$ is [2025]

Q 4 :

The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(−2, −3, 5) and D(1, −6, −7) is equal to [2023]

Q 5 :

The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to: [2023]

Q 6 :

Let the plane $x + 3 y - 2 z + 6 = 0$ meet the coordinate axes at the points A, B, C. If the orthocenter of the triangle ABC is $(α, β, \frac{6}{7})$ , then $98 {(α + β)}^{2}$ is equal to _____ . [2023]

Q 7 :

Let the direction cosines of two lines satisfy the equations $4 l + m - n = 0$ and $2 m n + 10 n l + 3 l m = 0 .$ Then the cosine of the acute angle between these lines is: [2026]

Q 8 :

Distance of the point $(α, β, γ)$ from Y-axis is :

Q 9 :

If the direction cosines of a line are $k$ , $k$ and $k$ , then :

Q 10 :

Under what condition do $(\frac{1}{\sqrt{2}}, \frac{1}{2}, k)$ represent direction cosines of a line?

Q 11 :

Any three numbers which are proportional to the direction cosines of a line, are called:

Q 12 :

Direction ratios of two lines are $a, b, c$ and $\frac{1}{b c}$ , $\frac{1}{c a}$ , $\frac{1}{a b}$ . The lines are :

Q 13 :

Direction ratios of the line represented by the equation $x = a y + b, z = c y + d$ are :

Q 14 :

If $α, β, γ$ are the angles that a line makes with the positive direction of $X$ , $Y$ , $Z$ axes, respectively, then the direction cosines of the line are :

Q 15 :

The distance of a point $P (a, b, c)$ from X-axis is :

Q 16 :

A line makes equal angles with coordinate axis. Direction cosines of this line are :

Q 17 :

The vector equation of the symmetrical form of equation of straight line $\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}$ is:

Q 18 :

The lines in a space which are neither intersecting nor parallel, are called:

Q 19 :

The two lines $x = a y + b, z = c y + d$ and $x = a' y + b', z = c' y + d'$ will be perpendicular, if and only if:

Q 20 :

A line makes angles of $45 °$ and $60 °$ with the positive axis of X and Y respectively. The angle made by the same line with the positive axis of Z, is:

Q 21 :

If $α, β, γ$ be the angles which a line makes with the coordinate axis, then:

Q 22 :

The equation of a line passing through the point $(- 3, 2, - 4)$ and equally inclined to the axis, are:

Q 23 :

The equation of straight line passing through the point $(a, b, c)$ and parallel to Z-axis is:

Q 24 :

The length of the perpendicular from point $(1, 2, 3)$ to the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ is:

Q 25 :

Equation of X-axis is:

Q 26 :

The equations of X-axis in space are:

Q 27 :

The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{4 - z}{k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$ are mutually perpendicular if the value of $k$ is:

Q 28 :

The shortest distance between the lines $x = y + 2 = 6 z - 6$ and $x + 1 = 2 y = - 12 z$ is:

Q 29 :

The distance of the plane $\vec{r} \cdot (\frac{2}{7} \hat{i} + \frac{3}{7} \hat{j} - \frac{6}{7} \hat{k}) = 1$ from the origin is:

Q 30 :

The locus represented by $x y + y z = 0$ is: