Distance of a Point from a Line

If the length of the perpendicular from the point $(1,1)$ to the line $ax-by+c=0$ be unity, show that $\frac{1}{c}+\frac{1}{a}-\frac{1}{b}=\frac{c}{2ab}$.

Show that the path of a moving point such that its distances from two lines $3x-2y=5$ and $3x+2y=5$ are equal is a straight line

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

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

Find the distance of the point $(1,2)$ from the straight line with slope $5$ and passing through the point of intersection of $x+2y=5$ and $x-3y=7$.

Find the perpendicular distance from the origin of the perpendicular from the point $(1,2)$ upon the straight line $x-\sqrt{3}y+4=0$.

Show that the product of perpendiculars on the line $\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$ from the points $(\pm \sqrt{a^2-b^2},0)$ is $b^2$.

What are the points on $x$ -axis whose perpendicular distance from the straight line $\frac{x}{a}+\frac{y}{b}=1$ is $a$?

Find the length of the perpendicular from the point $(4,-7)$ to the line joining the origin and the point of intersection of the lines $2x-3y+14=0$ and $5x+4y-7=0$.

Find the distance of the point of intersection of the lines $2x+3y=21$ and $3x-4y+11=0$ from the line $8x+6y+5=0$.

Show that the perpendiculars let fall from any point on the straight line $2x+11y-5=0$ upon the two straight lines $24x+7y=20$ and $4x-3y-2=0$ are equal to each other.

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

Find the distance of the point $(4,5)$ from the straight line $3x-5y+7=0$.

Find the equation of a line perpendicular to the line $\sqrt{3}x-y+5=0$ and at a distance of $3$ units from the origin.

If $p$ is the length of the perpendicular from the origin on the line $\frac{x}{a}+\frac{y}{b}=1$, then

If $\mathrm{L}$ is a variable line such that the algebraic sum of the distances of the points $(1,1)$, $(2,0)$ and $(0,2)$ from the line is equal to zero, then line $\mathrm{L}$ will always pass through the point  $(x,y)$ is

If the length of the perpendicular from the point $(1,1)$ to the line $ax-by+c=0$ be unity, show that $\frac{1}{c}+\frac{1}{a}-\frac{1}{b}=\frac{c}{2ab}$.

Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$