Line intersecting quadratic curve

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

4. Solve the simultaneous equations $$\begin{gathered} x + y = 2 \\ x ^ { 2 } + 2 y = 12 \end{gathered}$$

5. Solve the simultaneous equations $$\begin{array} { r } y - 3 x + 2 = 0 \\ y ^ { 2 } - x - 6 x ^ { 2 } = 0 \end{array}$$

6. (a) By eliminating \(y\) from the equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ show that $$x ^ { 2 } + 4 x - 8 = 0$$ (b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} y = x - 4 \\ 2 x ^ { 2 } - x y = 8 \end{gathered}$$ giving your answers in the form \(a \pm b \sqrt { } 3\), where \(a\) and \(b\) are integers.

1. Given the simultaneous equations

$$\begin{aligned} & 2 x + y = 1 \\ & x ^ { 2 } - 4 k y + 5 k = 0 \end{aligned}$$ where \(k\) is a non zero constant,

1. show that $$x ^ { 2 } + 8 k x + k = 0$$ Given that \(x ^ { 2 } + 8 k x + k = 0\) has equal roots,
2. find the value of \(k\).
3. For this value of \(k\), find the solution of the simultaneous equations.

Simplify

1. \(( 2 \sqrt { } 5 ) ^ { 2 }\)
2. \(\frac { \sqrt { } 2 } { 2 \sqrt { } 5 - 3 \sqrt { } 2 }\) giving your answer in the form \(a + \sqrt { } b\), where \(a\) and \(b\) are integers.

5. Solve the simultaneous equations $$\begin{gathered} y + 4 x + 1 = 0 \\ y ^ { 2 } + 5 x ^ { 2 } + 2 x = 0 \end{gathered}$$

5. (a) Show that eliminating \(y\) from the equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ produces the equation $$x ^ { 2 } + 8 x - 1 = 0$$ (b) Hence solve the simultaneous equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ giving your answers in the form \(a + b \sqrt { } 17\), where \(a\) and \(b\) are integers.

Solve the simultaneous equations $$\begin{gathered} y - 2 x - 4 = 0 \\ 4 x ^ { 2 } + y ^ { 2 } + 20 x = 0 \end{gathered}$$

5

1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7 .$$
2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

4 Solve the simultaneous equations $$x ^ { 2 } - 3 y + 11 = 0 , \quad 2 x - y + 1 = 0$$

8

1. Given that \(y = x ^ { 2 } - 5 x + 15\) and \(5 x - y = 10\), show that \(x ^ { 2 } - 10 x + 25 = 0\).
2. Find the discriminant of \(x ^ { 2 } - 10 x + 25\).
3. What can you deduce from the answer to part (ii) about the line \(5 x - y = 10\) and the curve \(y = x ^ { 2 } - 5 x + 15\) ?
4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
5. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 5 x + 15\) at the point \(( 5,15 )\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

7

1. Find the gradient of the line \(l\) which has equation \(x + 2 y = 4\).
2. Find the equation of the line parallel to \(l\) which passes through the point ( 6,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Solve the simultaneous equations $$y = x ^ { 2 } + x + 1 \quad \text { and } \quad x + 2 y = 4$$

5

1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7$$
2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

8 Find the points where the line \(y = 2 x - 3\) cuts the curve \(y = x ^ { 2 } - 4 x + 5\).

11

1. Show algebraically that the equation \(x ^ { 2 } - 6 x + 10 = 0\) has no real roots.
2. Solve algebraically the simultaneous equations \(y = x ^ { 2 } - 6 x + 10\) and \(y + 2 x = 7\).
3. Plot the graph of the function \(y = x ^ { 2 } - 6 x + 10\) on graph paper, taking \(1 \mathrm {~cm} = 1\) unit on each axis, with the \(x\) axis from 0 to 6 and the \(y\) axis from - 2 to 10 .
   On the same axes plot the line with equation \(y + 2 x = 7\) showing clearly where the line cuts the quadratic curve.
4. Explain why these \(x\) coordinates satisfy the equation \(x ^ { 2 } - 4 x + 3 = 0\). Plot a graph of the function \(y = x ^ { 2 } - 4 x + 3\) on the same axes to illustrate your answer.

7 Find the coordinates of the points where the line \(y = 3 x - 2\) cuts the curve \(y = x ^ { 2 } + 4 x - 8\).

9 Find the coordinates of the points where the curve \(y = x ^ { 2 } - 2 x - 8\) meets the line \(y = x + 2\).

3. (i) Solve the simultaneous equations $$\begin{aligned} & y = x ^ { 2 } - 6 x + 7 \\ & y = 2 x - 9 \end{aligned}$$ (ii) Hence, describe the geometrical relationship between the curve \(y = x ^ { 2 } - 6 x + 7\) and the straight line \(y = 2 x - 9\).

10. \includegraphics[max width=\textwidth, alt={}, center]{76efaf91-a6f3-4493-88d4-3654b023441d-3_646_773_986_477} The diagram shows the curve \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = 2 x + 1\). The curve and line intersect at the points \(P\) and \(Q\).

1. Using algebra, show that \(P\) has coordinates \(( 1,3 )\) and find the coordinates of \(Q\).
2. Find an equation for the tangent to the curve at \(P\).
3. Show that the tangent to the curve at \(Q\) has the equation \(y = 5 x - 11\).
4. Find the coordinates of the point where the tangent to the curve at \(P\) intersects the tangent to the curve at \(Q\).

4 Solve the simultaneous equations $$y = 2 ( x - 2 ) ^ { 2 } , \quad 3 x + y = 26$$

7 In this question you must show detailed reasoning.
Fig. 7 shows the curve \(y = 5 x - x ^ { 2 }\). \begin{figure}[h] \end{figure} The line \(y = 4 - k x\) crosses the curve \(y = 5 x - x ^ { 2 }\) on the \(x\)-axis and at one other point.
Determine the coordinates of this other point.

3

1. Determine, in terms of \(k\), the coordinates of the point where the lines with the following equations intersect. $$\begin{array} { r } x + y = k \\ 2 x - y = 1 \end{array}$$
2. Determine, in terms of \(k\), the coordinates of the points where the line \(\mathrm { x } + \mathrm { y } = \mathrm { k }\) crosses the curve \(y = x ^ { 2 } + k\).

10. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = 2 x + 1\). The curve and line intersect at the points \(P\) and \(Q\).

1. Using algebra, show that \(P\) has coordinates \(( 1,3 )\) and find the coordinates of \(Q\).
2. Find an equation for the tangent to the curve at \(P\).
3. Show that the tangent to the curve at \(Q\) has the equation \(y = 5 x - 11\).
4. Find the coordinates of the point where the tangent to the curve at \(P\) intersects the tangent to the curve at \(Q\).

The equation of a curve is \(y = \frac{1}{3}k^2x^2 - 2kx + 2\) and the equation of a line is \(y = kx + p\), where \(k\) and \(p\) are constants with \(0 < k < 1\).

1. It is given that one of the points of intersection of the curve and the line has coordinates \(\left(\frac{6}{5}, \frac{3}{5}\right)\). Find the values of \(k\) and \(p\), and find the coordinates of the other point of intersection. [7]
2. It is given instead that the line and the curve do not intersect. Find the set of possible values of \(p\). [3]