First slide

Planes in 3D

Question

The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, -1, 4) with the plane 5x - 4y - z- 1.
If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS is

Moderate

A

$\frac{1}{\sqrt{2}}$
B

$\sqrt{2}$
C

2
D

$2 \sqrt{2}$

Solution

Direction ratios of QR are 1 , 4, 1.
$Coordinates of P \equiv (\frac{4}{3}, \frac{1}{3}, \frac{13}{3})$
Direction ratios of PT are 2, 2, -1.
$Angle between QR and PT is 45^{\circ} .$
$And P T = 1$
$\Rightarrow PS = TS = \frac{1}{\sqrt{2}}$

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