Question

the locus of middle point of normal chords of the parabola y^2=4ax is

Answer 1

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Locus of the midpoint of any normal chords of y2=4ax is

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Solution

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Correct option is

B

x=a(y24a2+2+2a2y2)

Let AB be a normal chord where A≡(at12,2at1) and B≡(at22,2at2).

Let the midpoint of AB is P(h,k), then

2h=a(t12+t22)

=a[(t1+t2)2−2t1t2]

and 2k=2a(t1+t2)

We also have,

t2=−t1−t12

⇒t1+t2=t1−2 and t1t2=−t12−2

⇒ak=t1−2

⇒t1=−k2a

So, 2h=a[(−t12)2+2t12+4]

⇒h=a(t122+t12+2)

Thus, required locus is x=a(y24a2+2+2a2y2)