EXERCISE 10.3

The line through the points $\left(h,3\right)$ and $\left(4,1\right)$ intersects the line $7x-9y-19=0$ at right angle. Find the value of $h.$

Two lines passing through the point $\left(2,3\right)$ intersects each other at an angle of ${60}^{\mathrm{o}}.$ If slope of one line is $2,$ find equation of the other line.

Find the equation of the right bisector of the line segment joining the points $\left(3,4\right)$ and $\left(-1,2\right).$

Find the coordinates of the foot of perpendicular from the point $\left(-1, 3\right)$ to the line $3x-4y-16=0.$

The perpendicular from the origin to the line $y=mx+c$  meets it at the point $\left(-1,2\right).$ Find the values of $m$ and $c.$

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 $p^2+4q^2=k^2$.

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