. . Equation of tangent and normal to the parabola y ax (i) Equation of tangent in cartesian form Let P x y ) and Q x ) be two points on a parabola y ax - - Then, y = ax and y ax = a x Simplifying, y , the slope of the chord PQ . Thus = ) , represents the equation of the chord PQ .
When Q P , or y the chord becomes tangent at P . Thus the equation of tangent at x y ) is = ) where y is the slope of the tangent ... ( ) yy = ax ax yy ax = ax ax yy a x (ii) Equation of tangent in parametric form Equation of tangent at ( at at on the parabola is at = a x at yt at (iii) Equation of normal in cartesian form From ( ) the slope of normal is - y Therefore equation of the normal is = − a x ay ay = − y x y x xy ay = y xy ay x y ay (iv) Equation of normal in parametric form Equation of the normal at ( at at on the parabola is x at ay = at at at a xt = a at at xt at at Theorem . Three normals can be drawn to a parabola y ax from a given point, one of which is always real.
Fig. . P ( x ,y ) Q ( x ,y ) y = ax Q' Q'' - - Two Dimensional Analytical Geometry - II Proof ax is the given parabola. Let ( , ) α β be the given point.
Equation of the normal in parametric form is y = – tx at at ... ( ) If m is the slope of the normal then m = − . Therefore the equation ( ) becomes y = mx am am . Let it passes through ( , ) α β , then b = m am am a - am m β = which being a cubic equation in m , has three values of m .
Consequently three normals, in general, can be drawn from a point to the parabola, since complex roots of real equation, always occur in conjugate pairs and ( ) being an odd degree equation, it has atleast one real root. Hence atleast one normal to the parabola is real.