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Question

The equation of the locus of the point of intersection of two normals to the parabola y2=4ax which are perpendicular to each other is

A
y2=a(x3a)
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B
y2=a(x+3a)
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C
y2=a(x+2a)
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D
y2=a(x2a)
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Solution

The correct option is A y2=a(x3a)
Let P(x1,y1) be the point of intersectin of the two perpendicular normals at A(t1),B(t2) on the parabola y2=4ax
let t3 be the foot of the third normal through P.Equation of a normal at t to the parabola is y+xt=2at+at3
If this normal passes through P then y1+x1t=2at+at3at3+(2ax1)ty1=0(1)
Now t1,t2,t3 are the roots of (1). t1t2t3=y1a
Slope of the normal at t1 is t1 , slope of normal at t2 is t2
Normals at t1 and t2 are perpendicular (t1)(t2)=1t1t2t3=t3y1a=t3t3=y1a
t3 is a root of (1) a(y1a)3+(2ax1)(y1a)y1=0y31a2(2ax1)y1ay1=0
y21+a(2ax1)+a2=0y21=a(x13a) The locus of P is y2=a(x3a)

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