Question

The foot of the perpendicular from the point $(7, 14, 5)$ to the plane $2x + 4y - z = 2$ are

• A. (1, 2, 8)
• B. (3, 2, 8)
• C. (5, 10, 6)
• D. (9, 18, 4)

Answer 1

Answer: (A)

(1, 2, 8)

Answer 2

The correct answer is A:(1,2,8)
We know that the length of the perpendicular from the point $(x_1, y_1, z_1)$ to the plane
$ax + by + cz + d = 0$ is
$\frac{\left|ax_{1} + by_{1} +cz_{1} +d\right|}{\sqrt{a^{2} + b^{2} +c^{2}}}$
and the co-ordinate $\left(\alpha,\beta,\gamma\right)$ of the foot of the $\bot$ are given by
$\frac{\alpha-x_{1}}{a} = \frac{\beta-y_{1}}{b} = \frac{\gamma-z_{1}}{c}$
$= - \left(\frac{ax_{1} + by_{1} +cz_{1} +d}{a^{2}+b^{2} + c^{2}}\right)$ .......(1)
In the given ques,, $x_1 = 7, y_1 = 14, z_1 = 5,$
$a = 2 b = 4,c = -1, d = -2$
By putting these values in (1), we get
$\frac{\alpha -7}{2} = \frac{\beta -14}{4} = \frac{\gamma-5}{-1}= - \frac{63}{21}$
$\Rightarrow \, \alpha = 1 , \beta = 2$ and $\gamma = 8$
Hence, foot of $\bot$ is $(1, 2, 8)$