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Question
Equation of the line passing through the point (a cos3θ, a sin3θ) and perpendicular to the line x sec θ + y cosec θ = a is x cos θ – y sin θ = a sin 2θ.
Options
True
False
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Solution
This statement is False.
Explanation:
Equation of any line perpendicular to x sec θ + y cosec θ = a is
x cosec θ – y sec θ = k .......(i)
If equation (i) passes through (a cos3θ, a sin3θ) then
a cos3θ.cosec θ – a sin3θ.secθ = k
⇒ `(a cos^3 theta)/sintheta - (asin^3theta)/costheta` = k
∴ Required equation is
x cos θ – y sin θ = `(a cos^3 theta)/sintheta - (asin^3theta)/costheta`
⇒ `x/sintheta - y/costheta = a[(cos^4theta - sin^4theta)/(sintheta costheta)]`
⇒ `(xcostheta - ysintheta)/(sintheta costheta) = a[((cos^2theta + sin^2theta)(cos^2theta - sin^2theta))/(sintheta costheta)]`
⇒ x cos θ – y sin θ = a(cos2θ – sin2θ)
⇒ x cos θ – y sin θ = a cos 2θ.
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