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

Question

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 (A) √2 (B) 2 (C) √3 (D) 1

Tardigrade Admin · Accepted Answer

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,

d c_{s}

are equal &

d r_{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 = (3 - 2)^{2} + (0 - (- 1))^{2} + (3 - 2)^{2}

= 1^{2} + 1^{2} + 1^{2} = 3

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

$2$

$2$

$3$

$1$

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

2​

2

3​

1

$2$

$2$

$3$

$1$