. . Equations of tangent and normal at a point P on a given circle Tangent of a circle is a line which touches the circle at only one point and normal is a line perpendicular to the tangent and passing through the point of contact. Let P x y ( , and Q x be two points on the circle x gx fy Therefore, gx fy + = ...
( ) and x gx fy = ,,, ( ) ( ) ( ) gives g x f y )( )( g f g f = - Therefore, slope of PQ = − g f When Q P , the chord PQ becomes tangent at P Slope of tangent is − g f = − g f Hence, the equation of tangent is y = − ) ( g f . Simplifying, yy fy fy xx gx gx = xx yy gx fy gx fy ) = ...( ) Since ( , x y is a point on the circle, we have x gx fy Therefore, − gx fy gx fy ...( ) Hence, substituting ( ) in ( ), we get the equation of tangent at ( , x y as xx yy g x f y Fig. . P x y ( , Q x ′ Q ′′ Q C g f - - Hence, the equation of normal is = y f g ⇒ g = y f ⇒ g y ) + ) = y f x ) + ⇒ yx xy g y f x ) − ) = .
Remark ( ) The equation of tangent at x y ) to the circle x + y = a is xx yy ( ) The equation of normal at x y ) to the circle x + y = a is xy yx ( ) The normal passes through the centre of the circle.