The foot of the perpendicular drawn from A(1, | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The vector $\overrightarrow{AB}$ is parallel to the normal vector $\overrightarrow{N} = \hat{i} + 2\hat{j} + 2\hat{k}$ of the plane $x+2y+2z-5=0$.

Vedclass · Accepted Answer

$-7: 5$

Vedclass · Answer

(B) The vector $\overrightarrow{AB}$ is parallel to the normal vector $\overrightarrow{N} = \hat{i} + 2\hat{j} + 2\hat{k}$ of the plane $x+2y+2z-5=0$. Let $\frac{\alpha-1}{1} = \frac{\beta-2}{2} = \frac{\gamma-2}{2} = \lambda$. Then $\alpha = \lambda+1, \beta = 2\lambda+2, \gamma = 2\lambda+2$. Since $B$ lies on the plane $x+2y+2z-5=0$, we have $(\lambda+1) + 2(2\lambda+2) + 2(2\lambda+2) - 5 = 0$. $\lambda + 1 + 4\lambda + 4 + 4\lambda + 4 - 5 = 0 \Rightarrow 9\lambda + 4 = 0 \Rightarrow \lambda = -\frac{4}{9}$. Thus, $B = (1 - \frac{4}{9}, 2 - \frac{8}{9}, 2 - \frac{8}{9}) = (\frac{5}{9}, \frac{10}{9}, \frac{10}{9})$. Now, $\pi(A) = 1 + 2(2) + 2(2) + 5 = 1 + 4 + 4 + 5 = 14$. And $\pi(B) = \frac{5}{9} + 2(\frac{10}{9}) + 2(\frac{10}{9}) + 5 = \frac{5+20+20+45}{9} = \frac{90}{9} = 10$. Therefore, $-\pi(A) : \pi(B) = -14 : 10 = -7 : 5$.

The foot of the perpendicular drawn from $A(1, 2, 2)$ onto the plane $x+2y+2z-5=0$ is $B(\alpha, \beta, \gamma)$. If $\pi(x, y, z) \equiv x+2y+2z+5=0$ is a plane, then $-\pi(A) : \pi(B) =$ ?

Similar Questions

Find the equation of the line passing through $(1, 1, 1)$ and perpendicular to the plane $2x + 3y - z - 5 = 0$.

If the shortest distance between the lines $\frac{x - 1}{\alpha} = \frac{y + 1}{-1} = \frac{z}{1}, (\alpha \ne -1)$ and $x + y + z + 1 = 0 = 2x - y + z + 3$ is $\frac{1}{\sqrt{3}}$, then a value of $\alpha$ is

The image of the point $(-1, 3, 4)$ in the plane $x - 2y = 0$ is

What is the equation of the line passing through $(1, 2, 3)$ and perpendicular to the plane $3x + 4y - 5z = 6$?

The plane containing the line $\frac{x - 3}{2} = \frac{y + 2}{-1} = \frac{z - 1}{3}$ and also containing its projection on the plane $2x + 3y - z = 5$ contains which one of the following points?

Vedclass Test Series

Exam Paper Generator

Online Exam Module