Sketch the curve with equation $y=\left|\frac{4}{p-x}\right|$, where $p$ is a positive cons

Question

Sketch the curve with equation $y=\left|\frac{4}{p-x}\right|$, where $p$ is a positive constant. State, in terms of $p$, the equations of the asymptotes and the coordinates of any intersections with the coordinate axes. On the same diagram, sketch the line with equation $y=q(x-p)$, where $q$ is a positive constant.[4] Find, in terms of $p$ and $q$, the root of the equation $\left|\frac{4}{p-x}\right|=q(x-p)$.[3] Hence solve the inequality $\left|\frac{4}{p-x}\right|>q(x-p)$.[2]

Accepted Answer

(i) Asymptotes: $x=p$ and $y=0$; y-intercept: $\left(0,\frac{4}{p}\right)$; no x-intercept. (ii) $x=p+\frac{2}{\sqrt{q}}$. (iii) $x<p$ or $p<x<p+\frac{2}{\sqrt{q}}$.

Modulus Rational Function and Linear Intersection

Modulus Rational Function and Linear Intersection