Question

Find the equation to the tangent at $A$ to the circle $(x-a{)}^2+(y-b{)}^2=r^2$, where the radius through $A$ makes an angle $\alpha$ with the $x-axis$.

Answer 1

Coordinate of point A$=(a+r\cos \alpha ,b+\sin \alpha )$

Also since $\angle AOD=\alpha$

$\angle ADO=90-\alpha$

$\therefore$ Slope of tangent $=\tan (90-\alpha )$

$=-\cot \alpha$

$\therefore$ Equation of tangent,

$y-(b+r\sin \alpha )=-\cot \alpha (x-(a+r\cos \alpha )$

$y-b-r\sin \alpha =-\cot \alpha \left(x\right)+\cot \alpha (a+r\cos \alpha )$.