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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 intersection 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)

t1 t2 t3=y1a

Slope of the normal at t1 is t1

Slope of the normal at t2 is t2

Normals at t1 and t2 are perpendicular (t1)(t2)=1t1 t2=1t1 t2 t3=t3

y1a=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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