Line intersecting general conic

4 The equation of a curve is \(y ^ { 2 } + 2 x = 13\) and the equation of a line is \(2 y + x = k\), where \(k\) is a constant.

1. In the case where \(k = 8\), find the coordinates of the points of intersection of the line and the curve.
2. Find the value of \(k\) for which the line is a tangent to the curve.

4. Use algebra to solve the simultaneous equations $$\begin{array} { r } y - 3 x = 4 \\ x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 4 \end{array}$$ You must show all stages of your working.

2. Use algebra to solve the simultaneous equations $$\begin{aligned} x + y & = 5 \\ x ^ { 2 } + x + y ^ { 2 } & = 51 \end{aligned}$$ You must show all stages of your working.
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4. Solve the simultaneous equations $$\begin{aligned} & y = x - 2 , \\ & y ^ { 2 } + x ^ { 2 } = 10 . \end{aligned}$$

3. Solve the simultaneous equations $$\begin{gathered} x - 2 y - 1 = 0 \\ x ^ { 2 } + 4 y ^ { 2 } - 10 x + 9 = 0 \end{gathered}$$

5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 , \\ x ^ { 2 } + y ^ { 2 } = 29 . \end{gathered}$$

4. Solve the simultaneous equations $$\begin{aligned} x + y & = 2 \\ 4 y ^ { 2 } - x ^ { 2 } & = 11 \end{aligned}$$

11. \begin{figure}[h] \end{figure} The line \(y = x + 2\) meets the curve \(x ^ { 2 } + 4 y ^ { 2 } - 2 x = 35\) at the points \(A\) and \(B\) as shown in Figure 2.

1. Find the coordinates of \(A\) and the coordinates of \(B\).
2. Find the distance \(A B\) in the form \(r \sqrt { 2 }\) where \(r\) is a rational number.

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

5. Find the pairs of values \(( x , y )\) which satisfy the simultaneous equations $$\begin{aligned} & 3 x ^ { 2 } + y ^ { 2 } = 21 \\ & 5 x + y = 7 \end{aligned}$$

8 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 4 = 0\).

1. Find the centre and radius of the circle.
2. Find the coordinates of the points where the circle meets the line with equation \(y = 3 x + 4\).

4 Solve the simultaneous equations $$4 x ^ { 2 } + y ^ { 2 } = 10 , \quad 2 x - y = 4$$

7 Solve the simultaneous equations $$x + 2 y - 6 = 0 , \quad 2 x ^ { 2 } + y ^ { 2 } = 57 .$$

6 Solve the simultaneous equations $$2 x + y - 5 = 0 , \quad x ^ { 2 } - y ^ { 2 } = 3$$

3 Solve the simultaneous equations $$x ^ { 2 } + y ^ { 2 } = 34 , \quad 3 x - y + 4 = 0$$

7 In this question you must show detailed reasoning.
The points \(\mathrm { A } ( - 1,4 )\) and \(\mathrm { B } ( 7 , - 2 )\) are at opposite ends of a diameter of a circle.

1. Find the equation of the circle.
2. Find the coordinates of the points of intersection of the circle and the line \(y = 2 x + 5\).
3. Q is the point of intersection with the larger \(y\)-coordinate. Calculate the area of the triangle ABQ .

3. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
(b) Solve the simultaneous equations $$\begin{aligned} & x = 2 y - 2 \\ & x ^ { 2 } = y ^ { 2 } + 7 \end{aligned}$$

7. \begin{figure}[h] \end{figure} Fig. 1 shows the curve with equation \(y ^ { 2 } = 4 ( x - 2 )\) and the line with equation \(2 x - 3 y = 12\).
The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\).
2. Find, using algebra, the coordinates of \(P\) and \(Q\).
3. Show that \(\angle P A Q\) is a right angle.

8 In this question you must show detailed reasoning. Determine the coordinates of the point of intersection of the line with equation \(y = 2 x + 3\) and the curve with equation \(y ^ { 2 } - 4 x ^ { 2 } = 33\).

5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$

1 Solve the simultaneous equations. $$\begin{array} { r } x ^ { 2 } + 8 x + y ^ { 2 } = 84 \\ x - y = 10 \end{array}$$

1 Solve the simultaneous equations $$\begin{aligned} x ^ { 2 } + y ^ { 2 } & = 5 \\ y & = 2 x \end{aligned}$$

2 Solve the following simultaneous equations. $$x ^ { 2 } + 2 y ^ { 2 } = 36 \quad x + 2 y = 10$$

Solve the simultaneous equations $$y + 4x + 1 = 0$$ $$y^2 + 5x^2 + 2y = 0$$ [6]

Solve the simultaneous equations $$x - 2y = 1,$$ $$x^2 + y^2 = 29.$$ [6]