 # 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.

##### 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.

##### Suggested Handwritten and Video Solutions      ##### 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.



(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.



(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.



(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.



(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.



##### Suggested Handwritten and Video Solutions        ##### 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.



(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.



(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 $y=1$.



(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 $y=1$.



##### Suggested Handwritten and Video Solutions        ##### 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 $8$.

##### Suggested Handwritten and Video Solutions      ##### 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.



(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.



(ii)

Find the exact area bounded by curve $C$, the line $x=0$, $x=3$ and $x$-axis, giving your answer in terms of $a$.



(ii) Find the exact area bounded by curve $C$, the line $x=0$, $x=3$ and $x$-axis, giving your answer in terms of $a$.



##### Suggested Handwritten and Video Solutions        