Distance of a Point from a Line

Prove that the product of the lengths of the perpendiculars drawn from the points $\left(\sqrt{a^2-b^2},0\right)$ and $\left(-\sqrt{a^2-b^2},0\right)$ to the line $\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$ is $b^2.$

Find an equation of the line which is equidistant from parallel lines $9x+6y-7=0$ and $3x+2y+6=0.$

If sum of the perpendicular distances of a variable point $P\left(x,y\right)$ from the lines $x+y-5=0$ and $3x-2y+7=0$ is always $10.$ Show that $P$ must move on a line.

Find the perpendicular distance from the origin to the line joining the points $\left(\cos \theta ,\sin \theta \right)$ and $\left(\cos \varphi ,\sin \varphi \right).$

What are the points on the $y$-axis whose distance from the line $\frac{x}{3}+\frac{y}{4}=1$ is $4$ units.

If $p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $a$ and $b,$ then show that: $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$.

In the triangle $ABC$ with vertices $A\left(2,3\right),B\left(4,-1\right)$ and $C\left(1,2\right),$ find the equation and length of altitude from the vertex $A.$

If $p$ and $q$ are the lengths of perpendiculars from the origin to the lines $x\cos \theta -y\sin \theta =k\cos 2\theta$ and $x\sec \theta +y\mathrm{cosec}\theta =k,$ respectively, prove that $p^2+4q^2=k^2$.

Find the distance between parallel lines $l\left(x+y\right)+p=0$ and $l\left(x+y\right)-r=0$.

Find the distance between the parallel lines $15x+8y-34=0$ and $15x+8y+31=0$.

Find the points on the $x$-axis, whose distances from the line $\frac{x}{3}+\frac{y}{4}=1$ are $4$ units.

Find the distance of the point $\left(-1,1\right)$ from the line $12\left(x+6\right)=5\left(y-2\right).$