1.02q Use intersection points: of graphs to solve equations

9 The diagram shows a parabola \(P\) which has equation \(y = \frac { 1 } { 8 } x ^ { 2 }\), and another parabola \(Q\) which is the image of \(P\) under a reflection in the line \(y = x\). The parabolas \(P\) and \(Q\) intersect at the origin and again at a point \(A\).
The line \(L\) is a tangent to both \(P\) and \(Q\). \includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-5_1015_1089_623_479}

1. 1. Find the coordinates of the point \(A\).
   2. Write down an equation for \(Q\).
   3. Give a reason why the gradient of \(L\) must be - 1 .
2. 1. Given that the line \(y = - x + c\) intersects the parabola \(P\) at two distinct points, show that $$c > - 2$$
   2. Find the coordinates of the points at which the line \(L\) touches the parabolas \(P\) and \(Q\).
      (No credit will be given for solutions based on differentiation.)

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The line \(l\) has equation \(x - 2 y = c\) The points \(P\) and \(Q\) are the points of intersection of \(H\) and \(l\)

1. Determine, in terms of \(c\), the coordinates of \(P\) and the coordinates of \(Q\) The point \(R\) is the midpoint of \(P Q\)
2. Show that, as \(C\) varies, the coordinates of \(R\) satisfy the equation $$x y = - \frac { c ^ { 2 } } { a }$$ where \(a\) is a constant to be determined.

11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).

1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.

13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).

1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.

4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).

1. Use the graph to solve the following equations, showing your method clearly.
   (A) \(x + \frac { 1 } { x } = 4\) (B) \(2 x + \frac { 1 } { x } = 4\)
2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.

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 \(y ^ { 2 } - 4 x ^ { 2 } = 33\).

4

1. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided in the Printed Answer Booklet.
2. In this question you must show detailed reasoning. Find the exact coordinates of the points of intersection of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).

7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).

1. Show that the line does not meet the circle.
2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
   (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.

4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0\).

1. By completing the square, express this equation in the form $$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle, leaving your answer in surd form.
3. A line has equation \(y = 2 x\).
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation \(x ^ { 2 } - 4 x + 4 = 0\).
   2. Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, \(P\).
4. Prove that the point \(Q ( - 1,4 )\) lies inside the circle.

1 The straight line \(L\) has equation \(y = 3 x - 1\) and the curve \(C\) has equation $$y = ( x + 3 ) ( x - 1 )$$

1. Sketch on the same axes the line \(L\) and the curve \(C\), showing the values of the intercepts on the \(x\)-axis and the \(y\)-axis.
2. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation \(x ^ { 2 } - x - 2 = 0\).
3. Hence find the coordinates of the points of intersection of \(L\) and \(C\).

7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).

1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
2. The curve \(C\) and the line \(L\) intersect in two distinct points.
   1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
   2. Hence find the possible values of \(k\).

9 The diagram below shows the graphs of \(y = x ^ { 2 } - 4 x - 12\) and \(y = x + 2\) \includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-10_933_912_358_566} 9

1. Write down three inequalities which together describe the shaded region.
   9
2. Find the coordinates of the points \(A , B\) and \(C\).
   9
3. Find the exact area of the shaded region.
   Fully justify your answer.
   [0pt] [6 marks]

11 A curve, \(C\), passes through the point with coordinates \(( 1,6 )\) The gradient of \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 6 } ( x y ) ^ { 2 }$$ Show that \(C\) intersects the coordinate axes at exactly one point and state the coordinates of this point. Fully justify your answer.

10

1. Given that \(10 + 4 x - x ^ { 2 } \equiv p - ( x - q ) ^ { 2 }\), show that \(q = 2\) and find the value of \(p\).
2. Hence find the coordinates of all the points of intersection of the curve \(y = 10 + 4 x - x ^ { 2 }\) and the circle \(( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 25\).

7 A curve is given parametrically by \(x = 3 t , y = 1 + t ^ { 3 }\) where \(t\) is any real number.

1. Show that a cartesian equation for this curve is given by \(y = 1 + \frac { 1 } { 27 } x ^ { 3 }\). A second curve is given by \(y = x ^ { 2 } + 4 x - 19\).
2. Given that the curves intersect at the point \(( 3,2 )\), find the coordinates of all the other points of intersection between the two curves.

12 A particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

7 A curve is given parametrically by \(x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t\) where \(t\) is any real number.

1. Show that the equation of the normal to the curve at the point where \(t = 2\) can be written in the form \(2 x + 5 y = 30\).
2. Show that this normal does not meet the curve again.

\includegraphics{figure_9} The diagram shows the curves with equations \(y = x^3 - 3x + 3\) and \(y = 2x^3 - 4x^2 + 3\).

1. Find the \(x\)-coordinates of the points of intersection of the curves. [3]
2. Find the area of the shaded region. [4]

\includegraphics{figure_9} The diagram shows the curve \(y = (x - 2)^2\) and the line \(y + 2x = 7\), which intersect at points \(A\) and \(B\). Find the area of the shaded region. [8]

Functions \(f\) and \(g\) are defined by $$f : x \mapsto 2x + 5 \quad \text{for } x \in \mathbb{R},$$ $$g : x \mapsto \frac{8}{x - 3} \quad \text{for } x \in \mathbb{R}, x \neq 3.$$

1. Obtain expressions, in terms of \(x\), for \(f^{-1}(x)\) and \(g^{-1}(x)\), stating the value of \(x\) for which \(g^{-1}(x)\) is not defined. [4]
2. Sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) on the same diagram, making clear the relationship between the two graphs. [3]
3. Given that the equation \(fg(x) = 5 - kx\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\). [5]

The equation of a line is \(2y + x = k\), where \(k\) is a constant, and the equation of a curve is \(xy = 6\).

1. In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(AB\). [6]
2. Find the set of values of \(k\) for which the line \(2y + x = k\) intersects the curve \(xy = 6\) at two distinct points. [3]

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]