and finding the point of contact Let the line y mx + touch the circle x . The centre and radius of the circle are( , ) and a respectively. (i) Condition for a line to be tangent Then the perpendicular distance of the line y mx = from ( , ) is m m m | | This must be equal to radius .Therefore m or c m ) . Thus the condition for the line y mx + to be a tangent to the circle x is m ) .
(ii) Point of contact Let ( , x y be the the point of contact of y mx + with the circle x Then y = mx + . ... ( ) Equation of tangent at ( , x y is xx yy = a . yy = − xx ...
( ) Equations ( ) and ( ) represent the same line and hence the coefficients are proportional. So, y = − m y = a a m = − , c m = ± . Then the points of contact is either am m m or am m m Fig. .
( , ) P x y ( , C mx - - Two Dimensional Analytical Geometry - II Note The equation of tangent at P to a circle is y mx m ± . Theorem . From any point outside the circle x two tangents can be drawn. Proof Let P x y ( , be a point outside the circle.
The equation of the tangent is mx m ± . It passes through ( , x y . Therefore y = mx m ± mx = a m . Squaring both sides, we get mx = a m ( m x mx y a m = m mx y = .
This quadratic equation in m gives two values for m . These values give two tangents to the