SATHEE: 3D Geometry 3 Question 45

3D Geometry 3 Question 45

####45. A line with positive direction cosines passes through the point $P (2, - 1, 2)$ and makes equal angles with the coordinate axes. The line meets the plane $2 x + y + z = 9$ at point $Q$ . The length of the line segment $P Q$ equals

(2009)

(a) 1

(b) $\sqrt{2}$

(d) 2

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Answer:

Correct Answer: 45. (c)

Solution:

Since,

$l = m = n = \frac{1}{\sqrt{3}}$

$∴$ Equations of line are $\frac{x - 2}{1 / \sqrt{3}} = \frac{y + 1}{1 / \sqrt{3}} = \frac{z - 2}{1 / \sqrt{3}}$

$\Rightarrow$ $x - 2 = y + 1 = z - 2 = r$

[say]

$∴$ Any point on the line is

$Q \equiv (r + 2, r - 1, r + 2)$

$∵ Q$ lies on the plane $2 x + y + z = 9$

$∴ 2 (r + 2) + (r - 1) + (r + 2) = 9$

$\Rightarrow 4 r + 5 = 9 \Rightarrow r = 1$

$\Rightarrow Q (3, 0, 3)$

$∴ P Q = \sqrt{(3 - 2)^{2} + (0 + 1)^{2} + (3 - 2)^{2}} = \sqrt{3}$

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3D Geometry 3 Question 45

Answer:

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