Ans:
if the parabola is Y2= 4ax
take the focal chord which is easy for calculation e.x. LR (latus rectum)
then coordinates of extremities would be (a,2a) and (a,-2a)
equation of tangent of parabola at (a,2a) :
T=0 : 2ay = 2a(x+a)
y = x+a................................. (1)
equation of tangent of parabola at (a,-2a) :
T=0 -2ay = 2a (x+a)
y = -x-a .........................(2)
point of intersection of both tangents is (X1, Y1)
after solving eq1 and eq2 X1 = -a and Y1 = 0
so ( -a, 0)
eqn of normal of parabola at (a, 2a)
y = -x +3a ...............................(3)
eqn of normal of parabola at (a, -2a)
y = x -3a..............................(4)
so point of intersection of normal's : (X2, Y2)
after solving eq3 and eq4 X2= 3a and Y2 = 0
so we conclude... for y2= 4ax
Y1= Y2
similarly if u take Y2= -4aX then also you will get the same result..
in case of X2= 4aY and X2= -4 aY
you wil get X1 = X2
Thanks and Regards,
Ajay verma,
askIITians faculty,
IIT HYDERABAD
