C Maths June 89 Paper 1 Q16
(a) The parametric equations of a curve are x = a cos³ θ, y = a sin³ θ, where a is a positive constant and 0 < θ < π/2 (i) Show that dy/dx = tan θ (ii) The tangent to the curve at the point with parameter θ cuts the axes at S and T. Write down the equation of this tangent, and show that the distance, ST, is independent of θ.
(b) The parametric euqations of a curve are x = at, y = at², where a is a constant. The points P (ap, ap²) and Q (aq, aq²) lie on the curve. Find and simplify an expression, in terms of p and q, for the gradient of chord PQ. Deduce from your expression that the gradient of the tangent to the curve at p is 2p.
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Answer: (a)
(i) dy/dx = dy/dθ * dθ/dx
= (3a sin² θ cos θ) / (-3a cos² θ sin θ)
= -tan θ (shown)
(ii) Equation of tangent is thus
y = (-tan θ) x + c
Hence, y + (tan θ) x = c
substitute x = a cos³ θ, y = a sin³ θ
a sin³ θ + (tan θ)(a cos³ θ) = c
a sin³ θ + a sin θ cos² θ = c -----> tan θ = sin θ / cos θ
a sin θ (sin² θ + cos² θ) = c
c = a sin θ
Hence, equation of tangent is
y = (-tan θ) x + a sin θ
Let S and T be the x and y intercepts respectively.
When y = 0,
(tan θ) x = a sin θ
x = a cos θ
Hence, S = (a cos θ, 0)
When x = 0,
y = a sin θ
Hence, T = (0, a sin θ)
Hence,
Distance ST
= a, which is independent of θ (shown)
(b)
Gradient of chord PQ
= p + q
Gradient at P
= 2p
