The curve $C$ has equation $$y=ax^2+bx+c,$$ where $a,b,c$ are constants. It passes through

Question

The curve $C$ has equation $$y=ax^2+bx+c,$$ where $a,b,c$ are constants. It passes through $P\left(1,\frac92\right)$. The tangents to $C$ at the points where $x=1$ and $x=3$ have angles of inclination $\frac{\pi}{4}$ and $\frac{3\pi}{4}$, respectively. Angles of inclination are measured anticlockwise from the positive $x$-axis. Form three linear equations in $a,b,c$, and hence find their values.[5] The two tangents intersect at $T$. Find the exact area of the triangle bounded by the two tangents and the line $y=\frac92$.[3]

Accepted Answer

(i) $a=-\frac12,\ b=2,\ c=3$ (ii) $1$ square unit

A parabola determined by tangent inclinations

A parabola determined by tangent inclinations

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