Let the image of the point (1, 0, 7) in the line be the point . Then which one of the following points lis on the line passing through and making angles and with y-axis and z-axis respectively and an acute angle with x-axis? [2024]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
Let the image of the point (1, 0, 7) in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ be the point $(α, β, γ)$ . Then which one of the following points lies on the line passing through $(α, β, γ)$ and making angles $\frac{2 π}{3}$ and $\frac{3 π}{4}$ with y-axis and z-axis respectively and an acute angle with x-axis [2024]

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

2 $(1, - 2, 1 + \sqrt{2})$

3 $(1, 2, 1 - \sqrt{2})$

4 $(3, 4, 3 - 2 \sqrt{2})$

Ans.
(4)

Let $L_{1} : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} = λ$ (say)

$\Rightarrow x = λ, y = 2 λ + 1, z = 3 λ + 2$

Let the coordinates of M are $(λ, 2 λ + 1, 3 λ + 2)$ lie on the given line.

Let given point is P(1, 0, 7).

$∴$ Direction ratio of PM are $(λ - 1, 2 λ + 1, 3 λ - 5)$ PM is perpendicular the given line $L_{1}$ .

$∴ 1 (λ - 1) + 2 (2 λ + 1) + 3 (3 λ - 5) = 0$

$\Rightarrow λ - 1 + 4 λ + 2 + 9 λ - 15 = 0$

$\Rightarrow 14 λ - 14 = 0 \Rightarrow λ = 1$

$∴$ The coordinates of M are (1, 3, 5).

$∵$ M is mid point of PQ.

So, $\frac{α + 1}{2} = 1, \frac{β}{2} = 3$ and $\frac{γ + 7}{2} = 5$

$\Rightarrow α = 1, β = 6 and γ = 3$

$∴$ The image of point P(1, 0, 7) is Q(1, 6, 3).

Now the direction cosine of the line are

$m = \frac{\cos 2 π}{3} = \cos (π - \frac{π}{3}) = - \cos \frac{π}{3} = - \frac{1}{2}$

$n = \cos (\frac{3 π}{4}) = \cos (π - \frac{π}{4}) = - \frac{1}{\sqrt{2}}$

We know that, $l^{2} + m^{2} + n^{2} = 1$

$\Rightarrow l^{2} + \frac{1}{4} + \frac{1}{2} = 1$

$\Rightarrow l^{2} = 1 - \frac{1}{4} - \frac{1}{2} = \frac{4 - 1 - 2}{4} = \frac{1}{4} \Rightarrow l = \pm \frac{1}{2}$

$\Rightarrow l = \frac{1}{2}$ ( $∵$ Line make an acute angle with x-axis)

The equation of line passing through (1, 6, 3) with direction cosines $\frac{1}{2}, - \frac{1}{2}$ and $- \frac{1}{\sqrt{2}}$ is given by

$\vec{r} = \hat{i} + 6 \hat{j} + 3 \hat{k} + λ (\frac{1}{2} \hat{i} - \frac{1}{2} \hat{j} - \frac{1}{\sqrt{2}} \hat{k})$

$= (1 + \frac{λ}{2}) \hat{i} + (6 - \frac{λ}{2}) \hat{j} + (3 - \frac{λ}{\sqrt{2}}) \hat{k}$

Only option (4) satisfied the above equation for $λ$ = 4.

Want Us to Email you About Special Offers & Updates?