Practicing Success
If the distance of the point P(1, -2, 1) from the plane x + 2y -2z = α, where α > 0, is 5, then the foot of the perpendicular from P to the plane is |
$\left(\frac{8}{3}, \frac{4}{3}, \frac{-7}{3}\right)$ $\left(\frac{4}{3}, \frac{-4}{3}, \frac{1}{3}\right)$ $\left(\frac{1}{3}, \frac{2}{3}, \frac{10}{3}\right)$ $\left(\frac{2}{3}, -\frac{1}{3}, \frac{5}{2}\right)$ |
$\left(\frac{8}{3}, \frac{4}{3}, \frac{-7}{3}\right)$ |
We have, PM = 5 $⇒ \frac{|1-4-2-\alpha |}{\sqrt{1+4+4}}=5$ $⇒ |\alpha + 5| = 15 $ $⇒ \alpha + 5 = ± 15$ $⇒ \alpha = 10, -20 $ $⇒\alpha = 10$ $[∵ \alpha > 0 ]$ The equation of PM is $\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-1}{-2}$ The coordinates of M are given by $\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-1}{-2}=r$ So, the coordinates of M are (r + 1, 2r - 2, -2r + 1). As M lies on the plane $x + 2y - 2z = \alpha .$ $∴ r + 1 + 4r - 4 + 4r - 2 = 10 $ $[∵ \alpha = 10 ]$ $⇒ 9r = 15 ⇒ r = \frac{5}{3}$ Hence, the coordinates of M are $\left(\frac{8}{3}, \frac{4}{3}, \frac{-7}{3}\right)$ |