A curve $C$ has parametric equations $$x = t^2 + t, \quad y = t^2 - t, \quad t \in \mathbb{R}.$$ S

Question

A curve $C$ has parametric equations $$x = t^2 + t, \quad y = t^2 - t, \quad t \in \mathbb{R}.$$ Sketch the graph of $C$, indicating the coordinates of the points where the curve crosses either axis.[2] Find the coordinates of the point on $C$ where the tangent to $C$ is parallel to the line $5y = 4x - 20$.[3] Find the equation of the tangent to $C$ at the point $(0,2)$ and show algebraically that this tangent does not cut $C$ again.[5]

Accepted Answer

(i) Parabola $(x-y)^2=2(x+y)$, intercepts $(0,0),(0,2),(2,0)$. (ii) $\left(\frac{99}{4},\frac{63}{4}\right)$. (iii) $y=3x+2$; intersection equation $-2(t+1)^2=0$ gives only the tangent point (shown).

2024 JPJC Promo Q10

2024 JPJC Promo Q10

Solution:

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