9 An ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$
- Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes.
[0pt]
[2 marks] - Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(- 5 < k < 5\).
[0pt]
[5 marks] - The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { l } a
b \end{array} \right]\) to form another ellipse whose equation is \(9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c\). Find the values of the constants \(a , b\) and \(c\).
[0pt]
[5 marks] - Hence find an equation for each of the two tangents to the ellipse \(9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c\) that are parallel to the line \(y = x\).
[0pt]
[3 marks]
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