1.03e Complete the square: find centre and radius of circle

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.

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.

1 A circle with equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = k\) has a radius of 4 .

1. Find the coordinates of the centre of the circle.
2. Find the value of the constant \(k\).

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.

3. \begin{figure}[h] \end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.

1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
3. find the value of \(k\),
4. find the coordinates of \(D\).

16. \section*{Figure 3} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.

1. Find the coordinates of the centre of the circle.
2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.

4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 12 y + 12 = 0\).

1. By completing the square, 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. Show that the circle does not intersect the \(x\)-axis.
4. The line with equation \(x + y = 4\) intersects the circle at the points \(P\) and \(Q\).
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } + 3 x - 10 = 0$$
   2. Given that \(P\) has coordinates (2,2), find the coordinates of \(Q\).
   3. Hence find the coordinates of the midpoint of \(P Q\).

5 A circle with centre \(C\) has equation $$( x - 5 ) ^ { 2 } + ( y + 12 ) ^ { 2 } = 169$$

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the circle passes through the origin \(O\).
   2. Given that the circle also passes through the points \(( 10,0 )\) and \(( 0 , p )\), sketch the circle and find the value of \(p\).
3. The point \(A ( - 7 , - 7 )\) lies on the circle.
   1. Find the gradient of \(A C\).
   2. Hence find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2 Find the centre of the circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12\) Tick ( \(\checkmark\) ) one box.
(-2, -3) □
(-2, 3) □ \(( 2 , - 3 )\) □ \(( 2,3 )\) □

7 Three points \(A , B\) and \(C\) have coordinates \(A ( 8,17 ) , B ( 15,10 )\) and \(C ( - 2 , - 7 )\) 7

1. Show that angle \(A B C\) is a right angle.
   7
2. \(\quad A , B\) and \(C\) lie on a circle.
   7 (b) (i) Explain why \(A C\) is a diameter of the circle.
   7 (b) (ii) Determine whether the point \(D ( - 8 , - 2 )\) lies inside the circle, on the circle or outside the circle. Fully justify your answer.

9 The points \(P\) and \(Q\) have coordinates ( \(- 6,15\) ) and (12, 19) respectively. 9

1. 1. Find the coordinates of the midpoint of \(P Q\) 9
      1. (ii) Find the equation of the perpendicular bisector of \(P Q\) Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
         [0pt] [4 marks]
         9
   2. 1. A circle passes through the points \(P\) and \(Q\) The centre of the circle lies on the line with equation \(2 x - 5 y = - 30\) Find the equation of the circle. 9
   3. (ii) The circle intersects the coordinate axes at \(n\) points.
      State the value of \(n\) $$y = \sin x ^ { \circ }$$ for \(- 360 \leq x \leq 360\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-12_613_1552_532_246}

7 A circle has equation $$x ^ { 2 } + y ^ { 2 } - 6 x - 8 y = p$$ 7

1. 1. State the coordinates of the centre of the circle.
      7
      1. (ii) Find the radius of the circle in terms of \(p\).
         7
   2. The circle intersects the coordinate axes at exactly three points. Find the two possible values of \(p\).

14 Curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$$ 14

1. Curve \(C _ { 2 }\) is a reflection of \(C _ { 1 }\) in the line \(y = x\) Write down an equation of \(C _ { 2 }\) 14
2. Curve \(C _ { 3 }\) is a circle of radius 4 , centred at the origin.
   Describe a single transformation which maps \(C _ { 1 }\) onto \(C _ { 3 }\) 14
3. Curve \(C _ { 4 }\) is a translation of \(C _ { 1 }\) The positive \(x\)-axis and the positive \(y\)-axis are tangents to \(C _ { 4 }\) 14 (c) (i) Sketch the graphs of \(C _ { 1 }\) and \(C _ { 4 }\) on the axes opposite. Indicate the coordinates of the \(x\) and \(y\) intercepts on your graphs.
   [0pt] [2 marks]
   14 (c) (ii) Determine the translation vector.
   [0pt] [2 marks]
   14 (c) (iii) The line \(y = m x + c\) is a tangent to both \(C _ { 1 }\) and \(C _ { 4 }\) Find the value of \(m\)

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$

1. Determine the exact value of the eccentricity of \(E\) The points \(P ( 4 \cos \theta , 3 \sin \theta )\) and \(Q ( 4 \cos \theta , - 3 \sin \theta )\) lie on \(E\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l _ { 1 }\) is the normal to \(E\) at the point \(P\)
2. Use calculus to show that \(l _ { 1 }\) has equation $$4 x \sin \theta - 3 y \cos \theta = 7 \sin \theta \cos \theta$$ The line \(l _ { 2 }\) passes through the origin and the point \(Q\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(R\)
3. Determine, in simplest form, the coordinates of \(R\)
4. Hence show that, as \(\theta\) varies, \(R\) lies on an ellipse which has the same eccentricity as ellipse \(E\)

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x - 30 y + 209 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). The line \(L\) has equation \(y = m x + 1\), where \(m\) is a constant.
      Given that \(L\) is the tangent to \(C\) at the point \(P\),
2. show that $$2 m ^ { 2 } - 7 m - 22 = 0$$
3. Hence find the possible pairs of coordinates of \(P\).

5 A circle has equation \(x ^ { 2 } - 6 x + y ^ { 2 } - 4 y = 12\).

1. Show that the centre of the circle is at the point \(( 3,2 )\) and find the radius.
2. \(P Q\) is a diameter of the circle where \(P\) has coordinates \(( - 1 , - 1 )\). Find the equation of \(P Q\), giving your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
3. Another diameter of the circle passes through the point \(( 0,6 )\). Show that this diameter is perpendicular to \(P Q\).

5 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 5 . Find the coordinates of the centre of the circle and the value of \(k\).

1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).

1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
2. Find the coordinates of the midpoint of the chord \(A B\).

1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).

1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
2. Find the coordinates of the midpoint of the chord \(A B\).

The coordinates of three points \(A , B , C\) are \(( 4,6 ) , ( - 3,5 )\) and \(( 5 , - 1 )\) respectively. a) Show that \(B \widehat { A C }\) is a right angle.
b) A circle passes through all three points \(A , B , C\). Determine the equation of the circle.

The equation of a circle is \((x - a)^2 + (y - 3)^2 = 20\). The line \(y = \frac{1}{2}x + 6\) is a tangent to the circle at the point \(P\).

1. Show that one possible value of \(a\) is 4 and find the other possible value. [5]
2. For \(a = 4\), find the equation of the normal to the circle at \(P\). [4]
3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b). [4]

The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]

The equation of a circle is \((x-6)^2 + (y+a)^2 = 18\). The line with equation \(y = 2a - x\) is a tangent to the circle.

1. Find the two possible values of the constant \(a\). [5]
2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent. [3]

The coordinates of points \(A\), \(B\) and \(C\) are \((6, 4)\), \((p, 7)\) and \((14, 18)\) respectively, where \(p\) is a constant. The line \(AB\) is perpendicular to the line \(BC\).

1. Given that \(p < 10\), find the value of \(p\). [4]

A circle passes through the points \(A\), \(B\) and \(C\).

1. Find the equation of the circle. [3]
2. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(dx + ey + f = 0\), where \(d\), \(e\) and \(f\) are integers. [3]