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Tangents are drawn from any point on the line x+4a=0, to the parabola y^2=4ax . Prove that there chord of contact will subtends a right angle at the vertex.

Risshabh , 10 Years ago
Grade 11
anser 1 Answers
Arun

Last Activity: 4 Years ago

Parameterize parabola as (at²,2at) so tangent at t is ty=x+at²  
 
For points P(p) & Q(q) tangents are py=x+ap² & qy=x+aq² 
 
Solving for intersection gives y=a(p+q), x=apq
 
If this point lies on x+4a=0 then pq=−4 … (i)  
 
But if OP⊥OQ then 2ap/(ap²) * 2aq/(aq²) = −1 → pq=−4 which is true because of (i).
 
∴ ∠POQ=90°

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