A-Level H2 Math | 5 Essential Questions
Parametric Equations
Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. They are often used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the curves are called parametric curves or parametric surfaces.
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YI Tutorial Question 1
The parametric equations of a curve is given by
$x={{\sin }^{2}}t$, $y=3\cos t$ for $0\le t\le \pi $
(i)
Find the cartesian equation of this curve.
(i) Find the cartesian equation of this curve.
(ii)
Sketch the curve.
(ii) Sketch the curve.
(iii)
Find the value of $t$ at the point on curve whose tangent is parallel to the $y$-axis.
(iii) Find the value of $t$ at the point on curve whose tangent is parallel to the $y$-axis.
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2016 ACJC MYE Q4
The curve $C$ has parametric equation $x=3-2\cos \,t$, $y=-\tan \,t$ for $0<t<\frac{\pi }{2}$. Using differentiation, show that the curve $C$ does not have any stationary point.
[2]
(i)
A point $Q$ on $C$ has parameter $t=\frac{\pi }{6}$. Find the equation of the tangent at $Q$, leaving your answer in exact form.
[2]
(i) A point $Q$ on $C$ has parameter $t=\frac{\pi }{6}$. Find the equation of the tangent at $Q$, leaving your answer in exact form.
[2]
(ii)
Find the cartesian equation of the locus of the mid-point of $(3-2\cos \,t,\,-\tan \,t)$ and $(-3,0)$ as $t$ varies.
[3]
(ii) Find the cartesian equation of the locus of the mid-point of $(3-2\cos \,t,\,-\tan \,t)$ and $(-3,0)$ as $t$ varies.
[3]
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2018 TJC P2 Q1
A curve $C$ has parametric equations
$x=2{{e}^{t}}$, $y={{t}^{3}}-t$,
where $-1\le t\le \frac{3}{2}$.
(i)
Find $\frac{\text{d}y}{\text{d}x}$ in terms of $t$. Hence find the exact equations of the normals to the curve which are parallel to the $y$-axis.
[3]
(i) Find $\frac{\text{d}y}{\text{d}x}$ in terms of $t$. Hence find the exact equations of the normals to the curve which are parallel to the $y$-axis.
[3]
(ii)
Sketch the curve $C$.
(ii) Sketch the curve $C$.
(iii)
Find the equation of the tangent to $C$ at the point$\left( 2{{e}^{a}},{{a}^{3}}-a \right)$. Find the possible exact values of $a$ such that the tangent cuts the $y$-axis at $y=1$.
[4]
(iii) Find the equation of the tangent to $C$ at the point$\left( 2{{e}^{a}},{{a}^{3}}-a \right)$. Find the possible exact values of $a$ such that the tangent cuts the $y$-axis at $y=1$.
[4]
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2007 AJC P1 Q12 (b)
A curve is defined by the parametric equations
$x=a{{t}^{2}},\,y=2at.$
Show that the equation of the normal to the curve at the point $(a{{t}^{2}},\,2at)$ is at $y+tx=a{{t}^{3}}+2at$.
If the normal to the curve at the point $(a{{t}^{2}},\,2at)$ meets the curve again at the point $(a{{q}^{2}},\,2aq),$ show that ${{t}^{2}}+qt+2=0$.
Deduce that ${{q}^{2}}$ cannot be less than $8$.
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2018 EJC P1 Q3
The parametric equations of curve $C$, is given as $x=at$, $y=a{{t}^{3}}$, where $a$ is a positive constant.
(i)
The point $P$ on the curve has parameter $p$ and the tangent to the curve at point $P$ cuts the $y$-axis at $S$ and the $x$-axis at $T$. The point $M$ is the midpoint of $ST$. Find a Cartesian equation of the curve traced by $M$ as $p$ varies.
[5]
(i) The point $P$ on the curve has parameter $p$ and the tangent to the curve at point $P$ cuts the $y$-axis at $S$ and the $x$-axis at $T$. The point $M$ is the midpoint of $ST$. Find a Cartesian equation of the curve traced by $M$ as $p$ varies.
[5]
(ii)
Find the exact area bounded by curve $C$, the line $x=0$, $x=3$ and $x$-axis, giving your answer in terms of $a$.
[3]
(ii) Find the exact area bounded by curve $C$, the line $x=0$, $x=3$ and $x$-axis, giving your answer in terms of $a$.
[3]
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