TP and TQ are tangents to the parabola and the normals at P and Q meet at a point R on the curve Is, the center of the circle circumscribing the triangle TPQ lies on the parabola 2y2 = a(x – a). Or not ?

Class: 10

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True

False

answer is A.

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Detailed Solution

Let us assume that the point of contact of tangents be and
Let the center of the circle through triangle TPQ the (h, k)
∴ Point of intersection of tangents is T(at1t2, a(t1 + t2))
Equation of chord PQ:-
(t1 + t2)y = 2x + 2at1t2
2x – (t1 + t2)y + 2at1t2 = 0
Equation of circle through TPQ is;
   …(i)
It passes through T
⇒ λ = –a(t1t2 + 1)
Solving (i)
Now; comparing the general equation of circle with the center of the circle (h, k)
Substituting λ
     …(ii)
Also,
2k = 2at1 + 2at2 + (t1 + t2)λ
2k = 2at1 + 2at2 + (t1 + t2)(–a(t1t2 + 1))
2k = a{2(t1 + t2) – 3(t1 + t2)}
2k = –a(t1 + t2)
t1 + t2 =
Substituting in eqn.(i)
ah = 2k2 + a2
2k2 = a(h – a)
Now, generalizing the equation

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