2023 RVHS P1 Q10
A curve $C$ has parametric equations $x=\sqrt{4+{{t}^{2}}}$, $y={{t}^{2}}$, $t\ge 0$.
(i)
Find the equation of the tangent to curve $C$ at point $P\left( \sqrt{4+{{p}^{2}}},{{p}^{2}} \right)$.
[3]
(ii)
The tangent at $P$ meets the x-axis at point $Q$. Point $M$ is the midpoint of the line segment $PQ$. Find the cartesian equation of the curve traced out by $M$ as $p$ varies.
[3]
(iii)
Show that the area of the region bounded by $C$, the x-axis and the line $x=3$ is given by $\int_{a}^{b}{\frac{{{t}^{3}}}{\sqrt{4+{{t}^{2}}}}}\,\text{d}t$, where $a$ and $b$ are constants to be determined. Hence evaluate the integral to find the exact area of the region.
[4]
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