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Question

Show that if tangents be drawn from any point on the line x+4a=0, to the parabola y2=4ax. Prove that their chord of contact will subtend a right at the vertex.

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Solution

Solution:-
Equation to the chord of contact of the tangents drawn from a point (x1,y1), to the parabola y2=4ax is-
yy1=2a(x+x1)
Parameterize parabola as (at²,2at), so tangent at t is-
2aty=2a(x+at2)
ty=x+at2
For points P(p)&Q(q), tangents are-
py=x+ap2
x=pyap2(i)
qy=x+aq2
x=qyaq2(ii)
From eqn(i)&(ii), we have
pyap2=qyaq2
y(pq)=a(p2q2)
y=a(p+q){a2b2=(ab)(a+b)}
Substituting value of y in eqn(i), we have
x=p×(a(p+q))ap2
x=apq
If the point (apq,a(p+q)) lies on x+4a=0,
apq+4a=0
pq=4(iii)
But if OPOQ then 2ap(ap2)×2aq(aq2)=1pq=4 which is true because of eqn(iii).
POQ=90°
Hence, the chord of contact will subtend a right at the vertex.

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