The equation of the plane,passing through the midpoint of the line segment joining the points $P(1, 2, 5)$ and $Q(3, 4, 3)$ and perpendicular to it,is

The distance of the plane $\vec{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})$ from the origin is

The vector equation of the plane which is at a distance of $ \frac{3}{\sqrt{14}} $ from the origin and the normal vector from the origin is $ 2 \hat{i}-3 \hat{j}+\hat{k} $ is:

The combined equation for a pair of planes is $S \equiv 2 x^2-6 y^2-12 z^2+18 y z+2 z x+x y=0$. If one of the planes is parallel to $x+2 y-2 z=5$, then the acute angle between the planes $S=0$ is

The equation of the plane passing through the three points $(1, 1, 1)$,$(1, -1, 1)$,and $(-7, -3, -5)$ is:

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 $TRUE$?
$(A)$ $\alpha+\beta=2$
$(B)$ $\delta-\gamma=3$
$(C)$ $\delta+\beta=4$
$(D)$ $\alpha+\beta+\gamma=\delta$