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]

Question

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]

Smart Achievers · Accepted Answer

(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.

Solution

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

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

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

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

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Solution

Q. Let the image of the point (1, 0, 7) in the line x1=y–12=z–23 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 2π3 and 3π4 with y-axis and z-axis respectively and an acute angle with x-axis [2024]

1 (3,–4,3+22)

2 (1,–2,1+2)

3 (1,2,1–2)

4 (3,4,3–22)

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

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

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

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