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Question
Let C be a curve defined parametrically as \[x = a \cos^3 \theta, y = a \sin^3 \theta, 0 \leq \theta \leq \frac{\pi}{2}\] . Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).
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Solution
let point be p(x,y)
Given equation
x = a cos3θ
y = asin3θ
Slope of tangent = `"dy"/"dx"`
`"dx"/"dθ" = -3a cos^2θ sinθ`
`"dy"/"dθ" = 3a sin^2θ cosθ`
∴ `"dy"/"dx" = (dy/(dθ))/(dx/(dθ)) = (3a sin^2θ cosθ)/(-3a cos^2θ sinθ) = -tanθ` (∴ slope of tangent)
∴ Given two point (a ,0) and (0 , a)
∴ Slope of chord , m = `(y_2 - y_1)/(x_2 - x_1) = (a - 0)/(0 - a) = a/-a = -1`
∵ It is given that tangent to C is parallel to chord
∴ slope of tangent = slope of chord
⇒ -tanθ = -1
θ = `pi/4`
∴ `x = a cos^3θ`
x = `a cos^3(pi/4)`
`x = a(1/sqrt (2))^3 = a/(2 sqrt(2))`
`y = a sin^3θ`
`y = a sin^3 (pi/4)`
`y = a(1/sqrt (2))^3 = a/(2 sqrt(2))`
∴ Point P is `(a/(2 sqrt(2)) , a/(2 sqrt(2)))`
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