The line $PQ$ is tangent to the curve $y=10-\dfrac{18}{x^2}$ at $P$, where the $x$-coordinat

Question

The line $PQ$ is tangent to the curve $y=10-\dfrac{18}{x^2}$ at $P$, where the $x$-coordinate of $P$ is $3$. The tangent meets the $y$-axis at $Q$, and the curve meets the positive $x$-axis at $R$. Find the coordinates of $P$ and the equation of the tangent $PQ$.[3] Find the coordinates of $Q$ and $R$.[3] Hence find the exact area of the shaded region bounded by the curve, the tangent and the coordinate axes.[5]

Accepted Answer

(i) $P=(3,8)$, $y=\dfrac{4x}{3}+4$ (ii) $Q=(0,4)$, $R=\left(\dfrac3{\sqrt5},0\right)$ (iii) $12\sqrt5-18\text{ units}^2$

Rational curve with a tangent and the coordinate axes

Rational curve with a tangent and the coordinate axes

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