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Mathematics Question on Coordinate Geometry

Let the image of the point (1,0,7)(1, 0, 7) in the line x1=y−12=z−23\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} be the point (α,β,γ)(\alpha, \beta, \gamma). Then which one of the following points lies on the line passing through (α,β,γ)(\alpha, \beta, \gamma) and making angles 2π3\frac{2\pi}{3} and 3π4\frac{3\pi}{4} with the y-axis and z-axis respectively and an acute angle with the x-axis?

A

(1,−2,1+2)(1, -2, 1 + \sqrt{2})

B

(2,1,2−2)(2, 1, 2 - \sqrt{2})

C

(3,4,3−22)(3, 4, 3 - 2\sqrt{2})

D

(3,−4,3+22)(3, -4, 3 + 2\sqrt{2})

Answer

(3,4,3−22)(3, 4, 3 - 2\sqrt{2})

Explanation

Solution

To find the image of the point (1,0,7)(1, 0, 7) in the line r⃗1=y−12=z−23\frac{\vec{r}}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}, let us proceed with a step-by-step approach.

Equation of the Line
The line L1L_1 is given by:
r⃗1=y−12=z−23=λ\frac{\vec{r}}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} = \lambda
with direction vector b⃗=i^+2j^+3k^\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}.

Finding the Foot of Perpendicular (Point MM)
Let MM be the foot of the perpendicular from P(1,0,7)P(1, 0, 7) to L1L_1 with coordinates
(1+λ,1+2λ,2+3λ).(1 + \lambda, 1 + 2\lambda, 2 + 3\lambda).

The vector PM⃗\vec{PM} is:
PM⃗=(λ−1)i^+(1+2λ)j^+(3λ−5)k^.\vec{PM} = (\lambda - 1)\hat{i} + (1 + 2\lambda)\hat{j} + (3\lambda - 5)\hat{k}.

Condition of Perpendicularity
Since PM⃗\vec{PM} is perpendicular to the direction vector b⃗\vec{b}, we have:
PM⃗⋅b⃗=0.\vec{PM} \cdot \vec{b} = 0.
Expanding, we get:
(λ−1)+2(1+2λ)+3(3λ−5)=0.(\lambda - 1) + 2(1 + 2\lambda) + 3(3\lambda - 5) = 0.
Simplifying, we find:
14λ−14=0  ⟹  λ=1.14\lambda - 14 = 0 \implies \lambda = 1.

Thus, M=(2,3,5)M = (2, 3, 5).

Finding the Image Point Q(α,β,γ)Q(\alpha, \beta, \gamma)
Since MM is the midpoint of PP and QQ, we have:
Q=2M−P=(1,6,3).Q = 2M - P = (1, 6, 3).

Therefore, (α,β,γ)=(1,6,3)(\alpha, \beta, \gamma) = (1, 6, 3).

Verifying the Required Point on the Line
We need to find a point on the line passing through (1,6,3)(1, 6, 3) that makes angles π4\frac{\pi}{4} and π4\frac{\pi}{4} with the y-axis and z-axis, respectively, and an acute angle with the x-axis.
After verifying, the point that satisfies these conditions is:
Option (3): (3,4,3−23).\text{Option (3): } (3, 4, 3 - 2\sqrt{3}).

Thus, the correct answer is: (3,4,3−22)(3, 4, 3 - 2\sqrt{2})