Two curves intersecting

7. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 7 = 0 \\ & x ^ { 2 } + 2 x y - y ^ { 2 } = 7 \end{aligned}$$

9 Fig. 9 shows the curves \(y = \frac { 1 } { x + 2 }\) and \(y = x ^ { 2 } + 7 x + 7\). \begin{figure}[h] \end{figure}

1. Use Fig. 9 to estimate graphically the roots of the equation \(\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7\).
2. Show that the equation in part (i) may be simplified to \(x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0\). Find algebraically the exact roots of this equation.
3. The curve \(y = x ^ { 2 } + 7 x + 7\) is translated by \(\binom { 3 } { 0 }\).
   (A) Show graphically that the translated curve intersects the curve \(y = \frac { 1 } { x + 2 }\) at only one point. Estimate the coordinates of this point.
   (B) Find the equation of the translated curve, simplifying your answer.

1. In this question you must show detailed reasoning.

\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation \(y = 8 - 10 x + 6 x ^ { 2 } - x ^ { 3 }\) The curve \(C _ { 2 }\) has equation \(y = x ^ { 2 } - 12 x + 14\)

1. Verify that when \(x = 1\) the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect. The curves also intersect when \(x = k\).
   Given that \(k < 0\)
2. use algebra to find the exact value of \(k\).

11. \begin{figure}[h] \end{figure} The figure 5 shows part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { c c } C _ { 1 } : y = x ^ { 3 } - 2 x ^ { 2 } & x > 0 \\ C _ { 2 } : y = 9 - \frac { 5 } { 2 } ( x - 3 ) ^ { 2 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(P\) and \(Q\).
a. Verify that the point \(Q\) has coordinates \(( 3,9 )\) b. Use algebra to find the coordinates of the point \(P\).

7 Determine the points of intersection of the curve \(3 x y + x ^ { 2 } + 14 = 0\) and the line \(x + 2 y = 4\).

Find the coordinates of the points of intersection of the curve \(y = x^{\frac{2}{3}} - 1\) with the curve \(y = x^{\frac{1}{3}} + 1\). [4]