It is given that the straight line $l : y = mx + c$ is the tangent to the circle \(C : x^2 + y^2 =

Question

It is given that the straight line $l : y = mx + c$ is the tangent to the circle $C : x^2 + y^2 = r^2$ at point $P$. Using an algebraic method, show that $l$ is perpendicular to $OP$, i.e., tangent of a circle is perpendicular to the radius at the point of contact.

Accepted Answer

$P=\left(-\dfrac{mc}{1+m^2},\dfrac{c}{1+m^2}\right)$. For $m\ne0$, $m_{OP}=-\dfrac1m$; for $m=0$, $OP$ is vertical. $l\perp OP$ (shown).

Coordinate Geometry - Tangents and Normals to Circles

Coordinate Geometry - Tangents and Normals to Circles

Solution:

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