Let p1 and p2 denote the perpendicular distances from the origin to the lines x cos θ + y sin θ = 2a cos 4θ and x sec θ + y cosec θ = 4a cos 2θ, respectively. If mp1² + np2² = 4a², then which of the following is true?

Question

Accepted Answer

m = 1, n = 4. For the first line p1=2a*cos4t (already normalized), so p1^2=4a^2 cos^2(4t). For the second, p2=4a*cos2t/sqrt(sec^2 t+cosec^2 t)=2a*cos2t*sin2t=a*sin4t, so p2^2=a^2 sin^2(4t). Then m*p1^2+n*p2^2=4a^2 means 4m cos^2(4t)+n sin^2(4t)=4 for all t, giving m=1, n=4.

Let p1 and p2 denote the perpendicular distances from the origin to the lines x cos θ + y sin θ = 2a cos 4θ and x sec θ + y cosec θ = 4a cos 2θ, respectively. If mp1² + np2² = 4a², then which of the following is true?

Solution

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