Points A, B & C lie on the parabola y² = 4ax. The tangents to the parabola at A, B & C, taken in pairs,intersect at points P, Q & R. the ratio of the areas of the triangles ABC & PQR is 
where λ and μ are co-prime number then λ + μ is
Text Solution
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Sol. Equation of parabola is y 2 = 4ax .......(1)
Let A ≡ (at 12 , 2at 1 ) B ≡ (at 22 , 2at 2 ) , C ≡ (at 32 , 2at 3 )
Equation of the tangents to parabola (1) at A, B, C are
yt 1 = x + at 12 .......(2)
yt 2 = x + at 22 .......(3)
and yt 3 = x + at 32 .......(4)
Let the points of intersection of lines (2) , (3) be P; (3) , (4) be Q and (2) , (4) be R.
Then P ≡ (at 1 t 2 , a(t 1 + t 2 )) , Q ≡ (at 2 t 3 , a(t 2 + t 3 )), R ≡ (at 1 t 3 , a(t 1 + t 3 ))
Now area of Δ ABC,
Δ 1 = modulus of 
= modulus of 
. a. 2a 
= a 2 |(t 1 – t 2 ) (t 2 – t 3 ) (t 3 – t 1 )|
Area of Δ PQR
Δ 2 = modulus of 
= modulus of 
= modulus of 
[R 1 → R 1 – R 2 , R 2 → R 2 – R 3 ]
= modulus of (t 1 – t 3 ) (t 2 – t 1 ) (t 2 – t 3 )
= 
| (t 1 – t 2 ) (t 2 – t 3 ) (t 3 – t 1 ) |
Clearly 
= 
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