If the foot of the perpendicular drawn from the point $(1,0,3)$ on a line passing through $(\alpha, 7,1)$ is $\left(\frac{5}{3}, \frac{7}{3}, \frac{17}{3}\right)$, then $\alpha$ is equal to ____

JEE Main · 2020 · 07 Jan 2020 · Shift 2 · Three Dimensional Geometry

If the foot of the perpendicular drawn from the point

(1, 0, 3)

on a line passing through

(α, 7, 1)

(\frac{5}{3}, \frac{7}{3}, \frac{17}{3})

, then

α

is equal to ____

Answer: 4

Solution

Let

P = (1, 0, 3)

A = (α, 7, 1)

and the foot of the perpendicular

D = (\frac{5}{3}, \frac{7}{3}, \frac{17}{3})

. Since

D

is the foot,

A D ⊥ P D

, so

A D \cdot P D = 0

. With

A D = (\frac{5 - 3 α}{3}, - \frac{14}{3}, \frac{14}{3})

and

P D = (\frac{2}{3}, \frac{7}{3}, \frac{8}{3})

, the dot product gives

(5 - 3 α) (2) + (- 14) (7) + (14) (8) = 0

, i.e.

10 - 6 α - 98 + 112 = 0

, so

α = 4

JEE Main · 2020 · 07 Jan 2020 · Shift 2 · Integer answer

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