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  4. A line with positive direction cosines passes through the point P (2, - 1, 2) and makes equal angles with the coordinate axes. If the line meets the plane 2x + y + z = 9 at point Q, then the length PQ equals

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

Three Dimensional Geometry

Solution:

Point P is (2, - 1, 2)
Let this line meet at Q (h, k, w)
Direction ratio of this line is
(h - 2, k + 1, w - 2)
Since, $dc_s$ are equal & $dr_s$ are also equal,
So, h - 2 = k + 1 + w - 2
$\Rightarrow $ k = h - 3 and w = h
This line meets the plane
2x + y + z = 9 at Q, so,
2h + k + w = 9 or 2h + h - 3 + h = 9
$\Rightarrow $ 4h - 3 = 9 $\Rightarrow $ h = 3
and k = 0 and w = 3
Distance
$PQ = \sqrt{\left(3-2\right)^{2} + \left(0-\left(-1\right)\right)^{2} + \left(3-2\right)^{2}}$
$ = \sqrt{1^{2} + 1^{2}+ 1^{2}} = \sqrt{3} $