Two Dimensional Geometry jee mains questions | JEE Main Two Dimensional Geometry Previous Year Question

Q 21 :

If the angle between two lines is $\frac{π}{4}$ and slope of one of the lines is $\frac{1}{2}$ , then find the slope of the other line.

3 or $\frac{- 1}{3}$
2 or $\frac{- 1}{2}$
4 or $\frac{- 1}{4}$
3 or $- 3$

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(1)

$We know that the acute angle θ between two lines$ $with slopes m_{1} and m_{2} is given by$

$\tan θ =$ $| \frac{m_{2} - m_{1}}{1 + m_{1} m_{2}} |$ ...(i)

$Let$ $m_{1} = \frac{1}{2},$ $m_{2} = m$ $and$ $θ = \frac{π}{4}$

$Now, putting these values in (i), we get$

$\tan \frac{π}{4} =$ $| \frac{m - \frac{1}{2}}{1 + \frac{1}{2} m} |$ $\Rightarrow 1 =$ $| \frac{2 m - 1}{2 + m} |$

$which gives$ $\frac{2 m - 1}{2 + m} = 1$ $or$ $\frac{2 m - 1}{2 + m} = - 1$

$Therefore,$ $m = 3$ $or$ $m = - \frac{1}{3}$

$Hence, slope of the other line is$ $3$ $or$ $- \frac{1}{3} .$

Q 22 :

The angle between the lines $\sqrt{3} x - y - 2 = 0$ and $x - \sqrt{3} y + 1 = 0$ is

$90 °$
$60 °$
$45 °$
$30 °$

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(4)

$Let m_{1} and m_{2} are slopes of given lines$

$\Rightarrow$ $\tan θ = \frac{\sqrt{3} - \frac{1}{\sqrt{3}}}{1 + 1} = \frac{3 - 1}{2 \sqrt{3}} = \frac{1}{\sqrt{3}}$

$\Rightarrow$ $θ = \frac{π}{6} = 30 °$

Q 23 :

The angle between lines $\sqrt{3} x + y = 1$ and $x + \sqrt{3} y = 1$ is

$\frac{π}{6}$
$\frac{3 π}{4}$
$\frac{5 π}{2}$
$\frac{π}{3}$

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(1)

$Given equations of lines are$

$\sqrt{3} x + y = 1$ ...(i) and $x + \sqrt{3} y = 1$ ...(ii)

Let $m_{1}$ and $m_{2}$ be slopes of (i) and (ii), then

$m_{1} = - \sqrt{3}, m_{2} = - \frac{1}{\sqrt{3}}$

Let $θ$ be the angle between them, then

$\tan θ =$ $| \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$ $=$ $| \frac{- \sqrt{3} + \frac{1}{\sqrt{3}}}{1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}}} |$ $=$ $| \frac{- 3 + 1}{2 \sqrt{3}} |$ $= \frac{1}{\sqrt{3}}$ $\Rightarrow$ $θ = \frac{π}{6}$

Q 24 :

The angle between the lines $2 x - y + 3 = 0$ and $x + 2 y + 3 = 0$ is

$90 °$
$60 °$
$45 °$
$30 °$

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(1)

$Given equations of lines are$

$2 x - y + 3 = 0;$ $m_{1} = slope = 2$

$x + 2 y + 3 = 0;$ $m_{2} = slope = \frac{- 1}{2}$

So, $\tan θ =$ $| \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$ $=$ $| \frac{2 + \frac{1}{2}}{1 - 1} |$ $= \frac{5}{2 \times 0} = \frac{5}{0} = \infty$

$\Rightarrow$ $\tan θ = \tan \frac{π}{2} \Rightarrow θ = \frac{π}{2}$

Q 25 :

The equation of a straight line which is parallel to $x - 2 y + 8 = 0$ and passes through (0, 4) is:

$x - 2 y + 8 = 0$
$x - 2 y + 7 = 0$
$x - 2 y + 4 = 0$
$2 x + 2 y - 13 = 0$

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(1)

$The equation of a line parallel to the line x - 2 y + 8 = 0$ $is$ $x - 2 y + λ = 0$ ...(i)

$This passes through$ $(0, 4)$

$∴$ $0 - 2 (4) + λ = 0 \Rightarrow λ = 8$

$Putting$ $λ = 8$ $in (i), we get$

$x - 2 y + 8 = 0,$ which is the required equation.

Q 26 :

The equations of the lines through the point (3, 2) which makes an angle of $45 °$ with the line $x - 2 y = 3$ are

$3 x - y = 7$ and $x + 3 y = 9$
$x - 3 y = 7$ and $3 x + y = 9$
$x - y = 3$ and $x + y = 2$
$2 x + y = 7$ and $x - 2 y = 9$

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(1)

$Let the slope of the required line be m .$

$∴$ $\frac{y - 2}{x - 3} = m$ ...(i)

$Given line is$ $x - 2 y = 3 \Rightarrow y = \frac{1}{2} x - \frac{3}{2}$ ...(ii)

$Clearly, slope of the line is$ $\frac{1}{2}$

$It is given that the angle between (i) and (ii) is$ $45 °$

$∴$ $| \frac{m - \frac{1}{2}}{1 + \frac{1}{2} m} |$ $= \tan 45 °$ $\Rightarrow \frac{2 m - 1}{2 + m} = 1 or \frac{2 m - 1}{2 + m} = - 1$

$\Rightarrow$ $m = 3 or m = - \frac{1}{3}$

$∴$ $Required equation is \frac{y - 2}{x - 3} = 3 or \frac{y - 2}{x - 3} = - \frac{1}{3}$

$i.e.,$ $3 x - y - 7 = 0 or x + 3 y - 9 = 0 .$

Q 27 :

The equation of the line passing through the origin and the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{b} + \frac{y}{a} = 1$ is

$b x - a y = 0$
$x + y = 0$
$a x - b y = 0$
$x - y = 0$

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(4)

$Given equation of lines are$

$\frac{x}{a} + \frac{y}{b} = 1,$ $\frac{x}{b} + \frac{y}{a} = 1$

$or$ $b x + a y = a b$ ...(i)

$a x + b y = a b$ ...(ii)

$Solving (i) and (ii),$

$x = \frac{a b}{a + b},$ $y = \frac{a b}{a + b}$

$So equation of line passing through (0,0) is$

$y - 0 = \frac{\frac{a b}{a + b}}{\frac{a b}{a + b}} (x - 0) \Rightarrow y = x$

Q 28 :

Points (3, 3), ( $h$ , 0), (0, $k$ ) are collinear and $\frac{a}{h} + \frac{b}{k} = \frac{1}{3}$ . Then

a = 3, b = 2
a = 3, b = 3
a = 1, b = 1
a = 2, b = 2

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(3)

$Points (3, 3), (h, 0) and (0, k) are collinear, so one$ $point will lie on the line joining the other two points.$

$i.e. y - 0 = \frac{k - 0}{0 - h} (x - h)$

$∴$ $3 = - \frac{k}{h} (3 - h)$ $(∵ (3, 3) lies on the line)$

$\Rightarrow$ $\frac{3}{h} + \frac{3}{k} = 1 \Rightarrow \frac{1}{h} + \frac{1}{k} = \frac{1}{3}$

$Comparing with$ $\frac{a}{h} + \frac{b}{k} = \frac{1}{3},$ $we get$ $a = 1,$ $b = 1$

Q 29 :

A straight line passes through the points (5, 0) and (0, 3). The length of perpendicular from the point (4, 4) on the line is

$\frac{15}{\sqrt{34}}$
$\frac{\sqrt{17}}{2}$
$\frac{17}{2}$
$\sqrt{\frac{17}{2}}$

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(4)

$Equation of line passing through (5, 0) and (0, 3) is,$

$y - 3 = - \frac{3}{5} (x - 0)$

$\Rightarrow$ $5 y - 15 = - 3 x \Rightarrow 3 x + 5 y - 15 = 0$ ...(i)

$Now, perpendicular distance$

$=$ $| \frac{4 \times 3 + 4 \times 5 - 15}{\sqrt{9 + 25}} |$ $=$ $| \frac{12 + 20 - 15}{\sqrt{34}} |$ $=$ $| \frac{17}{\sqrt{34}} |$ $= \sqrt{\frac{17}{2}}$

Q 30 :

Find perpendicular distance of the line joining the points $(\cos θ, \sin θ)$ and $(\cos ϕ, \sin ϕ)$ from the origin.

$| \cos (\frac{θ - ϕ}{2}) |$
$| \cos (\frac{θ + ϕ}{2}) |$
$| \sin (\frac{θ + ϕ}{2}) |$
None of these

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(1)

$Let the points be A = (\cos θ, \sin θ) and B = (\cos ϕ, \sin ϕ)$

$Equation of line A B is$

$y - \sin θ = \frac{\sin ϕ - \sin θ}{\cos ϕ - \cos θ} (x - \cos θ)$

$\Rightarrow$ $y - \sin θ = \frac{2 \cos (\frac{θ + ϕ}{2}) \sin (\frac{ϕ - θ}{2})}{- 2 \sin (\frac{ϕ + θ}{2}) \sin (\frac{ϕ - θ}{2})} (x - \cos θ)$

$\Rightarrow$ $y \sin (\frac{θ + ϕ}{2}) - \sin θ \sin (\frac{θ + ϕ}{2}) = - x \cos (\frac{θ + ϕ}{2}) + \cos θ \cos (\frac{θ + ϕ}{2})$

$\Rightarrow$ $x \cos (\frac{θ + ϕ}{2}) + y \sin (\frac{θ + ϕ}{2}) - {\cos θ \cos (\frac{θ + ϕ}{2}) + \sin θ \sin (\frac{θ + ϕ}{2})} = 0$

$\Rightarrow$ $x \cos (\frac{θ + ϕ}{2}) + y \sin (\frac{θ + ϕ}{2}) - \cos (θ - \frac{θ + ϕ}{2}) = 0$

$\Rightarrow$ $x \cos (\frac{θ + ϕ}{2}) + y \sin (\frac{θ + ϕ}{2}) - \cos (\frac{θ - ϕ}{2}) = 0$

$Now, perpendicular distance of line from the origin is$

$= \frac{| 0 + 0 - \cos (\frac{θ - ϕ}{2}) |}{\sqrt{\cos^{2} (\frac{θ + ϕ}{2}) + \sin^{2} (\frac{θ + ϕ}{2})}} =$ $| \cos (\frac{θ - ϕ}{2}) |$

Q 31 :

Distance between lines $3 x + 4 y = 6$ and $6 x + 8 y = 15$ is

$\frac{7}{10}$
$\frac{3}{5}$
$\frac{1}{5}$
$\frac{3}{10}$

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(4)

$l_{1} : 3 x + 4 y = 6 & l_{2} : 2 (3 x + 4 y) = 15 are parallel$

$∴$ $Distance between l_{1} & l_{2} =$ $| \frac{(c_{1} - c_{2})}{\sqrt{{(3)}^{2} + {(4)}^{2}}} |$ $=$ $| \frac{(\frac{15}{2}) - 6}{\sqrt{25}} |$ $= \frac{\frac{3}{2}}{5} = \frac{3}{10}$

Q 32 :

If the equation of the locus of point equidistant from the points $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ is $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0$ , then $c =$

$a_{1}^{2} - a_{2}^{2} + b_{1}^{2} - b_{2}^{2}$
$\frac{1}{2} (a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2})$
$\sqrt{(a_{1}^{2} + b_{1}^{2} - a_{2}^{2} - b_{2}^{2})}$
$\frac{1}{2} (a_{2}^{2} + b_{2}^{2} - a_{1}^{2} - b_{1}^{2})$

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(4)

$Let α, β be the point of locus, equidistant from (a_{1}, b_{1}) and (a_{2}, b_{2}) is given by$

${(α - a_{1})}^{2} + {(β - b_{1})}^{2} = {(α - a_{2})}^{2} + {(β - b_{2})}^{2}$

$\Rightarrow$ $a_{1}^{2} + b_{1}^{2} - 2 a_{1} α - 2 b_{1} β - a_{2}^{2} - b_{2}^{2} + 2 a_{2} α + 2 b_{2} β = 0$

$\Rightarrow$ $2 (a_{2} - a_{1}) α + 2 (b_{2} - b_{1}) β + a_{1}^{2} + b_{1}^{2} - b_{2}^{2} - a_{2}^{2} = 0$

$\Rightarrow$ $(a_{2} - a_{1}) x + (b_{2} - b_{1}) y + \frac{1}{2} (a_{1}^{2} + b_{1}^{2} - a_{2}^{2} - b_{2}^{2}) = 0$

$\Rightarrow$ $c = - \frac{1}{2} [a_{1}^{2} + b_{1}^{2} - a_{2}^{2} - b_{2}^{2}] = \frac{1}{2} [a_{2}^{2} + b_{2}^{2} - a_{1}^{2} - b_{1}^{2}]$

Q 33 :

ABC is a triangle. G is the centroid. D is the midpoint of BC. If A = (2, 3) and G = (7, 5), then the point D is

$(\frac{9}{2}, 4)$
$(\frac{19}{2}, 6)$
$(\frac{11}{2}, \frac{11}{2})$
$(8, \frac{13}{2})$

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(2)

$Here (x_{1}, y_{1}) = (2, 3), (p, q) = (7, 5)$

$Centroid, G (p, q) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$

$\Rightarrow$ $\frac{x_{1} + x_{2} + x_{3}}{3} = 7, \frac{y_{1} + y_{2} + y_{3}}{3} = 5$

$\Rightarrow$ $2 + x_{2} + x_{3} = 21, 3 + y_{2} + y_{3} = 15$

$\Rightarrow$ $x_{2} + x_{3} = 19, y_{2} + y_{3} = 12$

$∴$ $D \equiv (\frac{19}{2}, \frac{12}{2}) = (\frac{19}{2}, 6)$

Q 34 :

The distance between the points $(a \cos α, a \sin α)$ and $(a \cos β, a \sin β)$ is

$2$ $| \sin (\frac{α - β}{2}) |$
$2$ $| a \sin (\frac{α - β}{2}) |$
$2$ $| a \cos (\frac{α - β}{2}) |$
$| a \cos (\frac{α - β}{2}) |$

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(2)

$The points are (a \cos α, a \sin α) and (a \cos β, a \sin β) .$

$So, d = \sqrt{a^{2} {(\cos α - \cos β)}^{2} + a^{2} {(\sin α - \sin β)}^{2}}$

$=$ $| a |$ $| \sqrt{1 + 1 - 2 \cos α \cos β - 2 \sin α \sin β} |$

$=$ $| a |$ $| \sqrt{4 \sin^{2} (\frac{α - β}{2})} |$ $= 2$ $| a \sin (\frac{α - β}{2}) |$

Q 35 :

The straight lines $x + 2 y - 9 = 0$ , $3 x + 5 y - 5 = 0$ and $a x + b y = 1$ are concurrent if the straight line $35 x - 22 y + 1 = 0$ passes through

$(a, b)$
$(b, a)$
$(a, - b)$
$(- a, b)$

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(1)

For $x + 2 y - 9 = 0$ , $3 x + 5 y - 5 = 0$ , $a x + b y - 1 = 0$

to be concurrent $| \begin{matrix} 1 & 2 & - 9 \\ 3 & 5 & - 5 \\ a & b & - 1 \end{matrix} |$ $= 0$

$\Rightarrow$ $35 a - 22 b + 1 = 0$ ...(i)

Comparing (i) with equation $35 x - 22 y + 1 = 0 .$

Hence line $35 x - 22 y + 1 = 0$ should pass through $(a, b)$ .

Q 36 :

The direction in which a straight line must be drawn through the point (1, 2) so that its point of intersection with the line $x + y = 4$ may be at a distance $\frac{\sqrt{6}}{3}$ from this point is

$30 °, 60 °$
$15 °, 75 °$
$∥ to x -axis$
$∥ to y -axis$

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(2)

$\frac{x - 1}{\cos θ} = \frac{y - 2}{\sin θ} = r = \frac{\sqrt{6}}{3} = \sqrt{\frac{2}{3}}$

The point $(r \cos θ + 1, r \sin θ + 2)$ lies on $x + y = 4$

$∴$ $r (\cos θ + \sin θ) = 1$

or $\cos θ + \sin θ = \sqrt{\frac{3}{2}}$ or $\frac{1}{\sqrt{2}} \cos θ + \frac{1}{\sqrt{2}} \sin θ = \frac{\sqrt{3}}{2}$

or $\cos (θ - 45 °) = \cos 30 °$

$∴$ $θ - 45 ° = \pm 30 °$

$∴$ $θ = 75 °, 15 ° .$

Q 37 :

The equations of the lines on which the perpendicular from the origin make $30 °$ angle with $x$ -axis and which form a triangle of area $\frac{50}{\sqrt{3}}$ with axes, are

$x + \sqrt{3} y \pm 10 = 0$
$\sqrt{3} x + y \pm 10 = 0$
$x \pm \sqrt{3} y - 10 = 0$
None of the above

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(2)

$Let p be the length of perpendicular from the origin$ $on the given line. Then its equation in normal form is given by$

$x \cos 30 ° + y \sin 30 ° = p or \sqrt{3} x + y = 2 p$

$This meets with the coordinate axes at$ $A (\frac{2 p}{\sqrt{3}}, 0) and B (0, 2 p) .$

$∴$ $Area of ∆ O A B = \frac{1}{2} (\frac{2 p}{\sqrt{3}}) (2 p) = \frac{2 p^{2}}{\sqrt{3}}$

$Also, it is given that$ $\frac{2 p^{2}}{\sqrt{3}} = \frac{50}{\sqrt{3}} \Rightarrow p = \pm 5$

$Hence, the required equations of the lines are$ $\sqrt{3} x + y \pm 10 = 0$

Q 38 :

The image of the point (2, 4) in the line $x + y - 10 = 0$ is

(4, 8)
(6, 5)
(6, 8)
(0, 10)

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(3)

$Let Q (α, β) be the image of point P (2, 4) in the line x + y - 10 = 0$

$The equation of line passing through$ $P (2, 4)$ $and perpendicular to$ $x + y - 10 = 0$ $is$ $(y - 4) = 1 (x - 2)$ $i.e.,$ $- x + y - 2 = 0$

$This meets with$ $x + y - 10 = 0$ $at$ $(4, 6)$ ,

$Then,$ $\frac{α + 2}{2} = 4, \frac{β + 4}{2} = 6$ $\Rightarrow$ $α = 6, β = 8$

Q 39 :

If the image of $(- \frac{7}{5}, - \frac{6}{5})$ in a line is (1, 2), then the equation of the line is

$3 x - y = 0$
$4 x - y = 0$
$3 x + 4 y = 1$
$4 x + 3 y = 1$

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(3)

$The required line is perpendicular bisector of the$ $line segment joining the two points.$

$Equation of line joining$ $(- \frac{7}{5}, - \frac{6}{5})$ $and$ $(1, 2)$ $is$

$\frac{y - 2}{x - 1} = \frac{\frac{- 6}{5} - 2}{\frac{- 7}{5} - 1}$ $\Rightarrow$ $4 x - 3 y + 2 = 0$

$A line perpendicular to this line is$ $3 x + 4 y + k = 0$ ...(i)

$Also, line (i) passes through the midpoint of$ $(- \frac{7}{5}, - \frac{6}{5})$ $and$ $(1, 2),$ $i.e.,$ $(\frac{- 1}{5}, \frac{2}{5})$

$∴$ $3 (- \frac{1}{5}) + 4 (\frac{2}{5}) + k = 0$ $\Rightarrow$ $k = - 1$

$So, the required line is$ $3 x + 4 y - 1 = 0$

Q 40 :

Let $A (1, 0), B (6, 2)$ and $C (\frac{3}{2}, 6)$ be the vertices of a triangle $A B C$ . If $P$ is a point inside the triangle $A B C$ such that the triangles $A P C, A P B$ and $B P C$ have equal areas, then the length of the line segment $P Q$ , where $Q$ is the point $(- \frac{7}{6}, - \frac{1}{3})$ , is ________.

(5)

$Since, ar (∆ P A B) = ar (∆ P A C) = ar (∆ P B C)$ $[Given]$

$∴$ $P will be centroid of ∆ A B C .$

$So, coordinates of P are$ $(\frac{17}{6}, \frac{8}{3}) .$

$Now,$ $P Q = \sqrt{{(\frac{24}{6})}^{2} + {(\frac{9}{3})}^{2}} = 5$

Q 41 :

The equation of the line parallel to the line $3 x - 4 y + 2 = 0$ and passing through $(- 2, 3)$ is

$3 x - 4 y + 18 = 0$
$3 x - 4 y - 18 = 0$
$3 x + 4 y + 18 = 0$
$3 x + 4 y - 18 = 0$

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(1)

$Given line is 3 x - 4 y + 2 = 0$ ...(i)

$Equation of the line parallel to the (i) is$ $3 x - 4 y + k = 0$ ...(ii)

$∴$ $Line (ii) passes through the point (- 2, 3) .$

$∴$ $3 (- 2) - (4 \times 3) + k = 0 \Rightarrow k = 18$

$∴$ $3 x - 4 y + 18 = 0$ $is the required equation of line.$

Q 42 :

Distance between two parallel lines $y = 2 x + 4$ and $y = 2 x - 1$ is

$5$
$5 \sqrt{5}$
$\sqrt{5}$
$\frac{1}{5}$

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(3)

Q 43 :

The equation of the line passing through the point $(- 3, 1)$ and bisecting the angle between co-ordinate axes is

$x + y + 2 = 0$
$- x + y + 2 = 0$
$x - y + 4 = 0$
$2 x + y + 5 = 0$

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(1)

Q 44 :

If the straight lines $2 x + 3 y - 1 = 0$ , $x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$ form a triangle with origin as orthocentre, then $(a, b)$ is given by

$(6, 4)$
$(- 3, 3)$
$(- 8, 8)$
$(0, 7)$

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(3)

Q 45 :

Lines are drawn parallel to the line $4 x - 3 y + 2 = 0$ , at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?

$(\frac{1}{4}, - \frac{1}{3})$
$(- \frac{1}{4}, - \frac{2}{3})$
$(- \frac{1}{4}, \frac{2}{3})$
$(\frac{1}{4}, \frac{1}{3})$

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(3)

The given line is $4 x - 3 y + 2 = 0$ .

The straight line parallel to the given line is $4 x - 3 y + c = 0$

Distance of this line from origin is $\frac{3}{5}$ .

$∴$ $| \frac{c}{\sqrt{16 + 9}} |$ $= \frac{3}{5} \Rightarrow$ $| \frac{c}{5} |$ $= \frac{3}{5} \Rightarrow c = \pm 3$

So, the parallel lines are $4 x - 3 y + 3 = 0$ or $4 x - 3 y - 3 = 0$ .

Now, point $(- \frac{1}{4}, \frac{2}{3})$ given in option (3) satisfies the equation, $4 x - 3 y + 3 = 0 .$

So, $(- \frac{1}{4}, \frac{2}{3})$ lies on the line $4 x - 3 y + 3 = 0$ .

Q 46 :

The equation of the line passing through the point $(- 3, 7)$ with slope zero is

$x = 7$
$y = 7$
$x = - 3$
$y = - 3$

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(2)

$Equation of line passing through (- 3, 7) and having slope m = 0 is,$

$y - 7 = 0 (x + 3)$ $\Rightarrow y - 7 = 0$ $\Rightarrow y = 7$

Q 47 :

If the straight lines $4 x + 6 y = 5$ and $6 x + k y = 3$ are parallel, then the value of $k$ is equal to

$\frac{- 2}{3}$
$8$
$9$
$10$

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(3)

$Since, the given lines are parallel.$

$So, their slopes must be equal.$

$∴$ $\frac{- 4}{6} = \frac{- 6}{k}$ $\Rightarrow k = \frac{- 6 \times 6}{- 4}$ $\Rightarrow k = 9$

Q 48 :

If the line $l_{1} : 3 y - 2 x = 3$ is the angular bisector of the lines $l_{2} : x - y + 1 = 0$ and $l_{3} : α x + β y + 17 = 0$ , then $α^{2} + β^{2} - α - β$ is equal to ______.

(348)

$Point of intersection of l_{1} : 3 y - 2 x = 3 and l_{2} : x - y + 1 = 0 is P (0, 1),$ $which lies on l_{3} : α x + β y + 17 = 0$

$β = - 17$

$Consider a random point Q (- 1, 0) on l_{2} : x - y + 1 = 0 .$ $Image of point Q about l_{1} : 2 x - 3 y + 3 = 0 is Q' (\frac{- 17}{13}, \frac{6}{13}) .$

$which can be calculated by formula,$ $\frac{x - (- 1)}{2} = \frac{y - 0}{- 3} = \frac{- 2 (- 2 + 3)}{13}$

$Now, Q' lies on l_{3} : α x + β y + 17 = 0$ $∴$ $α = 7$

$Now, α^{2} + β^{2} - α - β = 348$

Q 49 :

If the lines $5 x + k y = 7$ , $5 x - 2 y = 3$ and $3 x - y = 2$ are concurrent, then the value of $k$ is ______.

(2)

$Solving two equations x = 1, y = 1$

$Put in 5 x + k y = 7,$ $we get 5 + k = 7$ $\Rightarrow k = 7 - 5 = 2$

Q 50 :

If $λ : 1$ is the ratio in which the line joining the points (2, 2) and (4, 0) divides the segment joining the points (1, 2) and (4, 3), then the value of $3 λ$ is ______.

(1)

$The equation of the line joining the points (2, 2) and (4, 0) is$ $y - 2 = \frac{0 - 2}{4 - 2} (x - 2)$ $\Rightarrow x + y - 4 = 0$ $\dots \dots (i)$

$Suppose the line joining (2, 2) and (4, 0) divides the segment joining (1, 2) and (4, 3)$ $at the point P in the ratio of λ : 1 .$

$Then the coordinates of P are (\frac{4 λ + 1}{λ + 1}, \frac{3 λ + 2}{λ + 1})$

$Clearly, P lies on (i),$

$\Rightarrow$ $\frac{4 λ + 1}{λ + 1} + \frac{3 λ + 2}{λ + 1} - 4 = 0$

$\Rightarrow$ $λ = \frac{1}{3}$

Topic Question Set

Q 21 :

If the angle between two lines is $\frac{π}{4}$ and slope of one of the lines is $\frac{1}{2}$ , then find the slope of the other line.

Q 22 :

The angle between the lines $\sqrt{3} x - y - 2 = 0$ and $x - \sqrt{3} y + 1 = 0$ is

Q 23 :

The angle between lines $\sqrt{3} x + y = 1$ and $x + \sqrt{3} y = 1$ is

Q 24 :

The angle between the lines $2 x - y + 3 = 0$ and $x + 2 y + 3 = 0$ is

Q 25 :

The equation of a straight line which is parallel to $x - 2 y + 8 = 0$ and passes through (0, 4) is:

Q 26 :

The equations of the lines through the point (3, 2) which makes an angle of $45 °$ with the line $x - 2 y = 3$ are

Q 27 :

The equation of the line passing through the origin and the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{b} + \frac{y}{a} = 1$ is

Q 28 :

Points (3, 3), ( $h$ , 0), (0, $k$ ) are collinear and $\frac{a}{h} + \frac{b}{k} = \frac{1}{3}$ . Then

Q 29 :

A straight line passes through the points (5, 0) and (0, 3). The length of perpendicular from the point (4, 4) on the line is

Q 30 :

Find perpendicular distance of the line joining the points $(\cos θ, \sin θ)$ and $(\cos ϕ, \sin ϕ)$ from the origin.

Q 31 :

Distance between lines $3 x + 4 y = 6$ and $6 x + 8 y = 15$ is

Q 32 :

If the equation of the locus of point equidistant from the points $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ is $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0$ , then $c =$

Q 33 :

ABC is a triangle. G is the centroid. D is the midpoint of BC. If A = (2, 3) and G = (7, 5), then the point D is

Q 34 :

The distance between the points $(a \cos α, a \sin α)$ and $(a \cos β, a \sin β)$ is

Q 35 :

The straight lines $x + 2 y - 9 = 0$ , $3 x + 5 y - 5 = 0$ and $a x + b y = 1$ are concurrent if the straight line $35 x - 22 y + 1 = 0$ passes through

Q 36 :

The direction in which a straight line must be drawn through the point (1, 2) so that its point of intersection with the line $x + y = 4$ may be at a distance $\frac{\sqrt{6}}{3}$ from this point is

Q 37 :

The equations of the lines on which the perpendicular from the origin make $30 °$ angle with $x$ -axis and which form a triangle of area $\frac{50}{\sqrt{3}}$ with axes, are

Q 38 :

The image of the point (2, 4) in the line $x + y - 10 = 0$ is

Q 39 :

If the image of $(- \frac{7}{5}, - \frac{6}{5})$ in a line is (1, 2), then the equation of the line is

Q 41 :

The equation of the line parallel to the line $3 x - 4 y + 2 = 0$ and passing through $(- 2, 3)$ is

Q 42 :

Distance between two parallel lines $y = 2 x + 4$ and $y = 2 x - 1$ is

Q 43 :

The equation of the line passing through the point $(- 3, 1)$ and bisecting the angle between co-ordinate axes is

Q 44 :

If the straight lines $2 x + 3 y - 1 = 0$ , $x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$ form a triangle with origin as orthocentre, then $(a, b)$ is given by

Q 45 :

Lines are drawn parallel to the line $4 x - 3 y + 2 = 0$ , at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?

Q 46 :

The equation of the line passing through the point $(- 3, 7)$ with slope zero is

Q 47 :

If the straight lines $4 x + 6 y = 5$ and $6 x + k y = 3$ are parallel, then the value of $k$ is equal to

Q 48 :

If the line $l_{1} : 3 y - 2 x = 3$ is the angular bisector of the lines $l_{2} : x - y + 1 = 0$ and $l_{3} : α x + β y + 17 = 0$ , then $α^{2} + β^{2} - α - β$ is equal to ______.

Q 49 :

If the lines $5 x + k y = 7$ , $5 x - 2 y = 3$ and $3 x - y = 2$ are concurrent, then the value of $k$ is ______.

Q 50 :

If $λ : 1$ is the ratio in which the line joining the points (2, 2) and (4, 0) divides the segment joining the points (1, 2) and (4, 3), then the value of $3 λ$ is ______.

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Topic Question Set

Q 21 : If the angle between two lines is π4 and slope of one of the lines is 12, then find the slope of the other line.

Q 22 : The angle between the lines 3x-y-2=0 and x-3y+1=0 is

Q 23 : The angle between lines 3x+y=1 and x+3y=1 is

Q 24 : The angle between the lines 2x-y+3=0 and x+2y+3=0 is

Q 25 : The equation of a straight line which is parallel to x-2y+8=0 and passes through (0, 4) is:

Q 26 : The equations of the lines through the point (3, 2) which makes an angle of 45° with the line x-2y=3 are

Q 27 : The equation of the line passing through the origin and the point of intersection of the lines xa+yb=1 and xb+ya=1 is

Q 28 : Points (3, 3), (h, 0), (0, k) are collinear and ah+bk=13. Then

Q 29 : A straight line passes through the points (5, 0) and (0, 3). The length of perpendicular from the point (4, 4) on the line is

Q 30 : Find perpendicular distance of the line joining the points (cosθ, sinθ) and (cosϕ, sinϕ) from the origin.

Q 31 : Distance between lines 3x+4y=6 and 6x+8y=15 is

Q 32 : If the equation of the locus of point equidistant from the points (a1, b1) and (a2, b2) is (a1-a2)x+(b1-b2)y+c=0, then c=

Q 33 : ABC is a triangle. G is the centroid. D is the midpoint of BC. If A = (2, 3) and G = (7, 5), then the point D is

Q 34 : The distance between the points (acosα, asinα) and (acosβ, asinβ) is

Q 35 : The straight lines x+2y-9=0, 3x+5y-5=0 and ax+by=1 are concurrent if the straight line 35x-22y+1=0 passes through

Q 36 : The direction in which a straight line must be drawn through the point (1, 2) so that its point of intersection with the line x+y=4 may be at a distance 63 from this point is

Q 37 : The equations of the lines on which the perpendicular from the origin make 30° angle with x-axis and which form a triangle of area 503 with axes, are

Q 38 : The image of the point (2, 4) in the line x+y-10=0 is

Q 39 : If the image of (-75,-65) in a line is (1, 2), then the equation of the line is

Q 40 : Let A(1, 0), B(6, 2) and C(32,6) be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point (-76,-13), is ________.

Q 41 : The equation of the line parallel to the line 3x-4y+2=0 and passing through (-2,3) is

Q 42 : Distance between two parallel lines y=2x+4 and y=2x-1 is

Q 43 : The equation of the line passing through the point (-3,1) and bisecting the angle between co-ordinate axes is

Q 44 : If the straight lines 2x+3y-1=0, x+2y-1=0 and ax+by-1=0 form a triangle with origin as orthocentre, then (a,b) is given by

Q 45 : Lines are drawn parallel to the line 4x-3y+2=0, at a distance 35 from the origin. Then which one of the following points lies on any of these lines?

Q 46 : The equation of the line passing through the point (-3, 7) with slope zero is

Q 47 : If the straight lines 4x+6y=5 and 6x+ky=3 are parallel, then the value of k is equal to

Q 48 : If the line l1:3y-2x=3 is the angular bisector of the lines l2:x-y+1=0 and l3:αx+βy+17=0, then α2+β2-α-β is equal to ______.

Q 49 : If the lines 5x+ky=7, 5x-2y=3 and 3x-y=2 are concurrent, then the value of k is ______.

Q 50 : If λ:1 is the ratio in which the line joining the points (2, 2) and (4, 0) divides the segment joining the points (1, 2) and (4, 3), then the value of 3λ is ______.

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Q 21 :

If the angle between two lines is $\frac{π}{4}$ and slope of one of the lines is $\frac{1}{2}$ , then find the slope of the other line.

Q 22 :

The angle between the lines $\sqrt{3} x - y - 2 = 0$ and $x - \sqrt{3} y + 1 = 0$ is

Q 23 :

The angle between lines $\sqrt{3} x + y = 1$ and $x + \sqrt{3} y = 1$ is

Q 24 :

The angle between the lines $2 x - y + 3 = 0$ and $x + 2 y + 3 = 0$ is

Q 25 :

The equation of a straight line which is parallel to $x - 2 y + 8 = 0$ and passes through (0, 4) is:

Q 26 :

The equations of the lines through the point (3, 2) which makes an angle of $45 °$ with the line $x - 2 y = 3$ are

Q 27 :

The equation of the line passing through the origin and the point of intersection of the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{b} + \frac{y}{a} = 1$ is

Q 28 :

Points (3, 3), ( $h$ , 0), (0, $k$ ) are collinear and $\frac{a}{h} + \frac{b}{k} = \frac{1}{3}$ . Then

Q 29 :

A straight line passes through the points (5, 0) and (0, 3). The length of perpendicular from the point (4, 4) on the line is

Q 30 :

Find perpendicular distance of the line joining the points $(\cos θ, \sin θ)$ and $(\cos ϕ, \sin ϕ)$ from the origin.

Q 31 :

Distance between lines $3 x + 4 y = 6$ and $6 x + 8 y = 15$ is

Q 32 :

If the equation of the locus of point equidistant from the points $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ is $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0$ , then $c =$

Q 33 :

ABC is a triangle. G is the centroid. D is the midpoint of BC. If A = (2, 3) and G = (7, 5), then the point D is

Q 34 :

The distance between the points $(a \cos α, a \sin α)$ and $(a \cos β, a \sin β)$ is

Q 35 :

The straight lines $x + 2 y - 9 = 0$ , $3 x + 5 y - 5 = 0$ and $a x + b y = 1$ are concurrent if the straight line $35 x - 22 y + 1 = 0$ passes through

Q 36 :

The direction in which a straight line must be drawn through the point (1, 2) so that its point of intersection with the line $x + y = 4$ may be at a distance $\frac{\sqrt{6}}{3}$ from this point is

Q 37 :

The equations of the lines on which the perpendicular from the origin make $30 °$ angle with $x$ -axis and which form a triangle of area $\frac{50}{\sqrt{3}}$ with axes, are

Q 38 :

The image of the point (2, 4) in the line $x + y - 10 = 0$ is

Q 39 :

If the image of $(- \frac{7}{5}, - \frac{6}{5})$ in a line is (1, 2), then the equation of the line is

Q 41 :

The equation of the line parallel to the line $3 x - 4 y + 2 = 0$ and passing through $(- 2, 3)$ is

Q 42 :

Distance between two parallel lines $y = 2 x + 4$ and $y = 2 x - 1$ is

Q 43 :

The equation of the line passing through the point $(- 3, 1)$ and bisecting the angle between co-ordinate axes is

Q 44 :

If the straight lines $2 x + 3 y - 1 = 0$ , $x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$ form a triangle with origin as orthocentre, then $(a, b)$ is given by

Q 45 :

Lines are drawn parallel to the line $4 x - 3 y + 2 = 0$ , at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?

Q 46 :

The equation of the line passing through the point $(- 3, 7)$ with slope zero is

Q 47 :

If the straight lines $4 x + 6 y = 5$ and $6 x + k y = 3$ are parallel, then the value of $k$ is equal to

Q 48 :

If the line $l_{1} : 3 y - 2 x = 3$ is the angular bisector of the lines $l_{2} : x - y + 1 = 0$ and $l_{3} : α x + β y + 17 = 0$ , then $α^{2} + β^{2} - α - β$ is equal to ______.

Q 49 :

If the lines $5 x + k y = 7$ , $5 x - 2 y = 3$ and $3 x - y = 2$ are concurrent, then the value of $k$ is ______.

Q 50 :

If $λ : 1$ is the ratio in which the line joining the points (2, 2) and (4, 0) divides the segment joining the points (1, 2) and (4, 3), then the value of $3 λ$ is ______.