A turning point fixes a rational curve · source reference

Printed question

The curve $C$ has equation $y=\frac{ax+b}{x^2+c}$, where $a,b,c$ are constants. It is given that $x=2$ is an asymptote of $C$.

(i) State the value of $c$. [1]

(ii) Given also that $C$ has a turning point at $(6,\frac14)$, find the values of $a,b$. [4]

(iii) Sketch $C$, stating the equations of the asymptotes and the coordinates of the turning points. [You need not indicate the coordinates of intersections with the axes.] [3]

(iv) Hence find the range of values of $k$ for which the equation $kx^2-3x+10-4k=0$ has no real root. [2]

[2011/PJC/Promo/9]

Printed working

(i) $c=-4$.

(ii) Sub the point $(6,\frac14)$ and $c=-4$ into equation of curve,

$y=\frac{ax+b}{x^2-4}$.

$\frac14=\frac{6a+b}{6^2-4}$.

$6a+b=8$.

$b=8-6a$ — (1).

$\frac{dy}{dx}=\frac{(x^2-4)a-(ax+b)(2x)}{(x^2-4)^2}=\frac{ax^2-4a-2ax^2-2bx}{(x^2-4)^2}=\frac{-ax^2-2bx-4a}{(x^2-4)^2}$.

Since there is a turning point at $(6,\frac14)$,

$\Rightarrow\frac{dy}{dx}=0$ at $x=6$.

$\Rightarrow-a(6)^2-2b(6)-4a=0$.

$10a+3b=0$ — (2).

Solving simultaneously, $10a+3(8-6a)=0$, $8a=24$.

$\Rightarrow a=3,b=-10$.

(iii) Graph below: turning points $(\frac23,\frac94),(6,\frac14)$; asymptotes $x=-2,x=2,y=0$.

(iv) $kx^2-3x+10-4k=0$.

$kx^2-4k=3x-10$.

$k=\frac{3x-10}{x^2-4}$.

Hence, for $kx^2-3x+10-4k=0$ to have no real roots, the graph of $y=\frac{3x-10}{x^2-4}$ and $y=k$ must not intersect. From the sketch, they will not intersect for $\frac14<k<\frac94$.

Teacher notes

When $x=2$, denominator $=0\Rightarrow2^2+c=0$.

$C$ has turning pt at $(6,\frac14)\Rightarrow$ 1) $(6,\frac14)$ lies on $C$, so satisfies eqn of curve AND 2) At $x=6$, $\frac{dy}{dx}=0$.

Re-express equation such that one side of it is $\frac{ax+b}{x^2+c}$, where $a=3,b=-10,c=-4$.

Note: CANNOT use $b^2-4ac<0$ as question says “Hence” which means must use the curve sketched in the previous part.

Printed answer key

(i) $c=-4$. (ii) $a=3,b=-10$. (iii) Native graph above. (iv) $\frac14<k<\frac94$.

Source provenance

2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING. Question/answer booklet printed pages 1–14; solutions printed pages 1–28. Running metadata and repeated headings are transcribed in the topic metadata article.