+ to be a tangent to the conic sections

(i) parabola y ax Let ( , x y be the point on the parabola y ax . Then y ax ... ( ) Let y mx + be the tangent to the parabola ... ( ) Equation of tangent at x y ) to the parabola from .

. is yy a x ) . … ( ) Since ( ) and ( ) represent the same line, coefficients are proportional. y = m ax ⇒ y = m x m Then ( ) becomes, m   = a m ⇒ c m So the point of contact is m m   and the equation of tangent to parabola is y mx m The condition for the line y mx + to be tangent to the ellipse or hyperbola can be derived as follows in the same way as in the case of parabola.

(ii) ellipse x Condition for line mx to be the tangent to the ellipse is c a m , with the point of contact is a m and the equation of tangent is mx a m ± . (iii) Hyperbola x Condition for line mx + to be the tangent to the hyperbola is c a m , with the point of contact is a m and the equation of tangent is mx a m ± . Note ( ) In y mx a m ± , either y mx a m or y mx a m is the equation to the tangent of ellipse but not both. ( ) In y mx a m ± , either y mx a m or y mx a m is the equation to the tangent of hyperbola but not both.