Discussion questions

Sketch the following without the use of a graphic calculator, indicating clearly any axial intercepts, asymptotes, vertices/turning points and lines of symmetry.

(a) $x^2+x-3y=2$

(b) $y^2+y-3x=2$

(c) $a^2x^2+b^2y^2=r^2$, $a>b>0$ and $r>0$

(d) $x^2+4y^2+4x-8y+4=0$

(e) $36y^2=4x^2+1$

(f) $x^2-4y^2-2x-8y-39=0$

Sketch the curve $4x^2+y^2+2y-3=0$ and write down the equation(s) of its line(s) of symmetry and the maximum distance of the point $(0,5)$ to the curve.

Sketch the graph of $y=\dfrac2{x(x-\mu)}$, where $\mu$ is a positive constant, indicating clearly any axial intercept(s), asymptote(s) and coordinates of turning point(s).

Given $a=\dfrac\mu2$, deduce the range of values of $b$ in terms of $\mu$, such that the graphs of $b(x-a)^2+a^2y^2=a^2b$ and $y=\dfrac2{x(x-\mu)}$ intersect twice, where $a$ and $b$ are positive real numbers.

The curve $C$ has equation $y=\dfrac{ax^2+bx+c}{x-1}$, where $a$, $b$ and $c$ are non-zero constants.

(a) It is given that $C$ passes through the point $(3,23/2)$ and has a minimum point at $(2,10)$.

(i) Find the values of $a$, $b$ and $c$.

(ii) Sketch $C$, giving the coordinates of any turning points, points of intersection with the axes and the equations of any asymptotes.

(iii) Find the set of values of $m/k$, where $m$ and $k$ are positive constants, such that the curve with equation $\dfrac{x^2}{k^2}-\dfrac{(y-1)^2}{m^2}=1$ does not intersect $C$.

(b) In the case when $a=-1$ and $b=-1$, find the set of values of $c$, where $c\ne2$, such that $C$ has no stationary point.