3 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).
- Sketch the curve.
- Use calculus to find the equation of the tangent to the curve at A .
- Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65dd0efe-5c99-4814-b741-16e368c3469e-3_641_791_240_714}
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\caption{Fig. 10}
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A is the point with coordinates \(( 1,4 )\) on the curve \(y = 4 x ^ { 2 }\). B is the point with coordinates \(( 0,1 )\), as shown in Fig. 10. - The line through A and B intersects the curve again at the point C . Show that the coordinates of C are \(\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)\).
- Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = - 2 x - \frac { 1 } { 4 }\).
- The two tangents intersect at the point D . Find the \(y\)-coordinate of D .