The foot of the perpendicular from the point A \left(\right. 7 , 14 , 5 \left.\right) to the plane 2 x + 4 y

Engineering

Mathematics

Line of Greatest Slope

Question

The foot of the perpendicular from the point

A (7, 14, 5)

to the plane

2 x + 4 y - z = 2

is?

(3, 1, 8)

(1, 2, 8)

(3, - 3, 5)

(5, - 3, - 4)

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Let N be the foot of the perpendicular drawn from the point

A (7, 14, 5)

and perpendicular to the plane

2 x + 4 y - z = 2

Then, the equation of the line PN is

\frac{x - 7}{2} = \frac{y - 14}{4} = \frac{z - 5}{- 1} = λ

(say)

Let the coordinates of N be

N (2 λ + 7, 4 λ + 14, - λ + 5)

Since N lies on the plane

2 x + 4 y - z = 2

, so

2 (2 λ + 7) + 4 (4 λ + 14) - (- λ + 5) = 2

\Rightarrow 21 λ = - 63

\Rightarrow λ = - 3

∴

required foot of the perpendicular is

N (- 6 + 7, - 12 + 14, 3 + 5)

, i.e.,

N (1, 2, 8)

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