circle x Note ( ) If ( , x y is a point outside the circle, then both the tangents are real. ( ) If ( , x y is a point inside the circle, then both the tangents are imaginary. ( ) If ( , x y is a point on the circle, then both the tangents coincide. Example .
Find the equations of the tangent and normal to the circle at P ( , ) - . Equation of tangent to the circle at P x y ( , is xx yy That is, ( ) = y = Equation of normal is xy yx = That is, = . Example . If y is a tangent to the circle x , find c .
The condition for the line y mx + to be a tangent to the circle x is c m ) . Then, c = ± c = ± . Example . A road bridge over an irrigation canal have two semi circular vents each with a span of m and the supporting pillars of width m .
Use Fig. . to write the equations that represent the semi-verticular vents. Let O O be the centres of the two semi circular vents.
P x y ( , T T C m m O O m m - - First vent with centre O , ) and radius r = yields equation to first semicircle as = ⇒ x = , y > . Second vent with centre O ( , ) and radius r = yields equation to second vent as = ⇒ x = , y > .