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Question

Let two tangents are drawn to the curve y24(x+y)=3sinθ+4cosθ15, x,y,θR from the origin whose slopes are m1,m2. If the vertex of the curve is at maximum distance from the origin, then the value of 1m1m2 is


Solution

Given : y24(x+y)=3sinθ+4cosθ15
y24y+4=4x+3sinθ+4cosθ11(y2)2=4(x113sinθ4cosθ4)(y2)2=4(xλ)

This is a parabola whose vertex is (λ,2)
Now, 
λ=113sinθ4cosθ4
We know that, 
3sinθ+4cosθ[5,5]λ[32,4]
Vertex is maximum when λ=4, so the parabola becomes
(y2)2=4(x4)
Now, equation of tangent of slope m is
y2=m(x4)+1m

As the tangents are passing through origin, so putting (0,0) in the equation, we get
2=4m+1m4m22m1=0
The roots of this equation are m1,m2, then using product of roots, we get
m1m2=141m1m2=4

Mathematics

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