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).

Answer 1

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)))`