to (i) an ellipse and (ii) a hyperbola from any external point on the plane. ( ) The locus of the point of intersection of perpendicular tangents to (i) the parabola y ax is x = − (the directrix). (ii) the ellipse x = is x ( called the director circle of ellipse). (iii) the hyperbola x = is x (called director circle of hyperbola).
Example . Find the equations of tangent and normal to the parabola x at ( , Equation of parabola is x −+ + = x + = - ... ( ) Let X = x Y Equation ( ) takes the standard form X = - Y Equation of tangent is XX = − Y Y At ( , X = = −−= − ; Y Therefore, the equation of tangent at ( , is x + = - - - x + = −+ . + = .
Slope of tangent at( , is - , so slope of normal at ( , is Therefore, the equation of normal at ( , is given by y + = x - y + = x - = . Example . Find the equations of tangent and normal to the ellipse x when θ = . - - Equation of ellipse is = = a = , b = a = , b = Equation of tangent at θ = p is = = = .
Equation of normal is = That is = = . Aliter At, θ = p , ( cos , sin ) θ = = ( , ) ∴ Equation of tangent at θ = is same at ( , ) . Equation of tangent in cartesian form is