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If the tangents and the normals at the extremities of a focal chord intersect at (x1,y1) and (x2,y2) respectively then: a) x1=x2 b) x1=y2 c) y1=y2 d) x2=y1 Also explain your answer...

If the tangents and the normals at the extremities of a focal chord intersect at (x1,y1) and (x2,y2) respectively then:


a) x1=x2


b) x1=y2


c) y1=y2


d) x2=y1


Also explain your answer...

Grade:upto college level

1 Answers

Ajay Verma
askIITians Faculty 33 Points
10 years ago
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

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