The image of a point $A(3, 8)$ in the line $x + 3y - 7 = 0$ is

The equation of the straight line passing through the point of intersection of the lines $5x - 6y - 1 = 0$ and $3x + 2y + 5 = 0$ and perpendicular to the line $3x - 5y + 11 = 0$ is

Find the equation of a line which passes through $(2 \cos^3 \theta, 2 \sin^3 \theta)$ and is perpendicular to the line $x \cos \theta - y \sin \theta = 2 \cos 2 \theta$.

Suppose $P(x,y)$ lies on $\sqrt{3} x-y+2=0$ or $\sqrt{3} x+y-2=0$ and is at a distance of $5$ units from their point of intersection. Then the distance from $(0,0)$ to the foot of the perpendicular of $P$ onto the $y$-axis is

Let $\alpha$ and $\beta$ be integers satisfying $0 < \beta < \alpha$. Let $P(\alpha, \beta)$ be a point. Let $Q$ be the reflection of $P$ in the line $y = x$,$R$ be the reflection of $Q$ in the $y$-axis,$S$ be the reflection of $R$ in the $x$-axis,and $T$ be the reflection of $S$ in the $y$-axis. If the area of the convex pentagon $PQRST$ is $187 \ sq. \ units$,then the value of $\alpha + \beta^2$ is:

The polar coordinates of $P$ are $\left(2, \frac{\pi}{6}\right)$. If $Q$ is the image of $P$ about the $X$-axis,then the polar coordinates of $Q$ are...