1.02c Simultaneous equations: two variables by elimination and substitution

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 .

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 \(y = x ^ { 2 } + k\).

1 The point \(A\) has coordinates \(( 11,2 )\) and the point \(B\) has coordinates \(( - 1 , - 1 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation $$x - 4 y = 3$$
2. The line with equation \(3 x + 5 y = 26\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).

3 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 12 x - 6 y + 20 = 0\).

1. By completing the square, express the equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle.
3. The line with equation \(y = x + 4\) intersects the circle at the points \(P\) and \(Q\).
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } - 5 x + 6 = 0$$
   2. Find the coordinates of \(P\) and \(Q\).

3

1. 1. Express \(x ^ { 2 } - 4 x + 9\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence, or otherwise, state the coordinates of the minimum point of the curve with equation \(y = x ^ { 2 } - 4 x + 9\).
2. The line \(L\) has equation \(y + 2 x = 12\) and the curve \(C\) has equation \(y = x ^ { 2 } - 4 x + 9\).
   1. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation $$x ^ { 2 } - 2 x - 3 = 0$$
   2. Hence find the coordinates of the points of intersection of \(L\) and \(C\).

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0\).

1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. The point \(D\) has coordinates (7, -2).
   1. Verify that the point \(D\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(D\), giving your answer in the form \(m x + n y = p\), where \(m , n\) and \(p\) are integers.
4. 1. A line has equation \(y = k x\). Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$
   2. Find the values of \(k\) for which the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (d)(ii).

7

1. 1. Express \(4 - 10 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = 4 - 10 x - x ^ { 2 }\).
2. The curve \(C\) has equation \(y = 4 - 10 x - x ^ { 2 }\) and the line \(L\) has equation \(y = k ( 4 x - 13 )\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
   2. Given that the curve \(C\) and the line \(L\) intersect in two distinct points, show that $$4 k ^ { 2 } + 33 k + 29 > 0$$
   3. Solve the inequality \(4 k ^ { 2 } + 33 k + 29 > 0\).

6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.

1. The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
2. The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
3. Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
4. Hence determine the possible values of the width of the garden. \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
   1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
   2. Write down:
      1. the coordinates of \(C\);
      2. the radius of the circle.
      3. Sketch the circle.
      4. A line has equation \(y = k x + 6\), where \(k\) is a constant.
         1. Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
         2. The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
         3. Hence find the values of \(k\) for which the line is a tangent to the circle.

1 The point \(A\) has coordinates \(( - 3,2 )\) and the point \(B\) has coordinates \(( 7 , k )\).
The line \(A B\) has equation \(3 x + 5 y = 1\).

1. 1. Show that \(k = - 4\).
   2. Hence find the coordinates of the midpoint of \(A B\).
2. Find the gradient of \(A B\).
3. A line which passes through the point \(A\) is perpendicular to the line \(A B\). Find an equation of this line, giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
4. The line \(A B\), with equation \(3 x + 5 y = 1\), intersects the line \(5 x + 8 y = 4\) at the point \(C\). Find the coordinates of \(C\).

1 The point \(A\) has coordinates \(( 6,5 )\) and the point \(B\) has coordinates \(( 2 , - 1 )\).

1. Find the coordinates of the midpoint of \(A B\).
2. Show that \(A B\) has length \(k \sqrt { 13 }\), where \(k\) is an integer.
3. 1. Find the gradient of the line \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x - 2 y = 8\).
4. The line \(A B\) intersects the line with equation \(2 x + y = 10\) at the point \(C\). Find the coordinates of \(C\).

1 The point \(A\) has coordinates \(( 1,7 )\) and the point \(B\) has coordinates \(( 5,1 )\).

1. 1. Find the gradient of the line \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x + 2 y = 17\).
2. The line \(A B\) intersects the line with equation \(x - 4 y = 8\) at the point \(C\). Find the coordinates of \(C\).
3. Find an equation of the line through \(A\) which is perpendicular to \(A B\).

1 The trapezium \(A B C D\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-2_298_591_557_737} The line \(A B\) has equation \(2 x + 3 y = 14\) and \(D C\) is parallel to \(A B\).

1. Find the gradient of \(A B\).
2. The point \(D\) has coordinates \(( 3,7 )\).
   1. Find an equation of the line \(D C\).
   2. The angle \(B A D\) is a right angle. Find an equation of the line \(A D\), giving your answer in the form \(m x + n y + p = 0\), where \(m , n\) and \(p\) are integers.
3. The line \(B C\) has equation \(5 y - x = 6\). Find the coordinates of \(B\).

1 The line \(A B\) has equation \(7 x + 3 y = 13\).

1. Find the gradient of \(A B\).
2. The point \(C\) has coordinates \(( - 1,3 )\).
   1. Find an equation of the line which passes through the point \(C\) and which is parallel to \(A B\).
   2. The point \(\left( 1 \frac { 1 } { 2 } , - 1 \right)\) is the mid-point of \(A C\). Find the coordinates of the point \(A\).
3. The line \(A B\) intersects the line with equation \(3 x + 2 y = 12\) at the point \(B\). Find the coordinates of \(B\).

2 The line \(A B\) has equation \(4 x - 3 y = 7\).

1. 1. Find the gradient of \(A B\).
   2. Find an equation of the straight line that is parallel to \(A B\) and which passes through the point \(C ( 3 , - 5 )\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
2. The line \(A B\) intersects the line with equation \(3 x - 2 y = 4\) at the point \(D\). Find the coordinates of \(D\).
3. The point \(E\) with coordinates \(( k - 2,2 k - 3 )\) lies on the line \(A B\). Find the value of the constant \(k\).

5

1. 1. Express \(x ^ { 2 } - 3 x + 5\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = x ^ { 2 } - 3 x + 5\).
2. The curve \(C\) with equation \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = x + 5\) intersect at the point \(A ( 0,5 )\) and at the point \(B\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-4_471_707_653_676}
   1. Find the coordinates of the point \(B\).
   2. Find \(\int \left( x ^ { 2 } - 3 x + 5 \right) \mathrm { d } x\).
   3. Find the area of the shaded region \(R\) bounded by the curve \(C\) and the line segment \(A B\).

1 The line \(A B\) has equation \(3 x - 4 y + 5 = 0\).

1. The point with coordinates \(( p , p + 2 )\) lies on the line \(A B\). Find the value of the constant \(p\).
2. Find the gradient of \(A B\).
3. The point \(A\) has coordinates ( 1,2 ). The point \(C ( - 5 , k )\) is such that \(A C\) is perpendicular to \(A B\). Find the value of \(k\).
4. The line \(A B\) intersects the line with equation \(2 x - 5 y = 6\) at the point \(D\). Find the coordinates of \(D\).

5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x + 3$$ where \(c\) and \(d\) are integers.

1. Given that \(x + 3\) is a factor of \(\mathrm { p } ( x )\), show that $$3 c - d = 8$$
2. The remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\) is 65 . Obtain a further equation in \(c\) and \(d\).
3. Use the equations from parts (a) and (b) to find the value of \(c\) and the value of \(d\). [3 marks]

1 The line \(A B\) has equation \(5 x + 3 y + 3 = 0\).

1. The line \(A B\) is parallel to the line with equation \(y = m x + 7\). Find the value of \(m\).
2. The line \(A B\) intersects the line with equation \(3 x - 2 y + 17 = 0\) at the point \(B\). Find the coordinates of \(B\).
3. The point with coordinates \(( 2 k + 3,4 - 3 k )\) lies on the line \(A B\). Find the value of \(k\).
   [0pt] [2 marks]

3.

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

5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
3. Find the exact coordinates of the mid-point of \(A C\).

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.

4. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,

1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
2. form a quadratic inequality in \(x\).
3. by solving your inequalities, find the set of possible values of \(x\).

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 \(Q\).

9. The curve \(C\) has the equation \(y = x ^ { 2 } + 2 x + 4\).

1. Express \(x ^ { 2 } + 2 x + 4\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point of \(C\). The straight line \(l\) has the equation \(x + y = 8\).
2. Sketch \(l\) and \(C\) on the same set of axes.
3. Find the coordinates of the points where \(I\) and \(C\) intersect.