If $a, b, c$ are the intercepts made by the plane passing through the point $(1, 2, 3)$ parallel to the plane $3x + 4y - 5z = 0$ on the $X, Y, Z$-axes respectively,then $3a + b + 5c =$

If a plane cuts off intercepts $OA = a, OB = b, OC = c$ from the coordinate axes,then the area of the triangle $ABC$ is:

$A$ plane $\pi$ passing through the point $(1,1,1)$ is perpendicular to the line joining the points $(6,3,2)$ and $(1,-4,-9)$. If $ax+by+cz-23=0$ is the equation of the plane $\pi$, then $a+b-c=$

Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^2+\beta^2+\gamma^2 \neq 0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are $TRUE$?
$(A)$ $\alpha+\beta=2$
$(B)$ $\delta-\gamma=3$
$(C)$ $\delta+\beta=4$
$(D)$ $\alpha+\beta+\gamma=\delta$

The equation of the plane passing through the origin and perpendicular to the planes $x+2y-z=1$ and $3x-4y+z=5$ is:

$A$ plane $\pi_1$ passing through the point $3 \hat{i}-7 \hat{j}+5 \hat{k}$ is perpendicular to the vector $\hat{i}+2 \hat{j}-2 \hat{k}$ and another plane $\pi_2$ passing through the point $2 \hat{i}+7 \hat{j}-8 \hat{k}$ is perpendicular to the vector $3 \hat{i}+2 \hat{j}+6 \hat{k}$. If $p_1$ and $p_2$ are the perpendicular distances from the origin to the planes $\pi_1$ and $\pi_2$ respectively, then $p_1-p_2=$