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

2 The equation of a circle is \(x ^ { 2 } - 12 x + y ^ { 2 } + 8 y + 3 = 0\).

1. Find the radius of the circle.
2. State the coordinates of the centre of the circle.

8 The curves \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) and \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) crosses the \(x\)-axis at D and the \(y\)-axis at C . \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-7_826_819_520_255}

1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
2. Determine the coordinates of A, B, C and D.
3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(\mathrm { y } = \mathrm { mx } + c\), where \(m\) and \(c\) are constants to be determined.
4. Points A , B , C and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\), \(b\) and \(r ^ { 2 }\) are constants to be determined.

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 \(Q\).

5 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x + 6 y = 11\).

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. The point \(O\) has coordinates \(( 0,0 )\).
   1. Find the length of CO .
   2. Hence determine whether the point \(O\) lies inside or outside the circle, giving a reason for your answer.

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 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 4 x + 12 y + 15 = 0\).

1. Find:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. Explain why the circle lies entirely below the \(x\)-axis.
3. The point \(P\) with coordinates \(( 5 , k )\) lies outside the circle.
   1. Show that \(P C ^ { 2 } = k ^ { 2 } + 12 k + 45\).
   2. Hence show that \(k ^ { 2 } + 12 k + 20 > 0\).
   3. Find the possible values of \(k\).

6 A circle has centre \(C ( - 3,1 )\) and radius \(\sqrt { 13 }\).

1. 1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
   2. Hence find the equation of the circle in the form $$x ^ { 2 } + y ^ { 2 } + m x + n y + p = 0$$ where \(m , n\) and \(p\) are integers.
2. The circle cuts the \(y\)-axis at the points \(A\) and \(B\). Find the distance \(A B\).
3. 1. Verify that the point \(D ( - 5 , - 2 )\) lies on the circle.
   2. Find the gradient of \(C D\).
   3. Hence find an equation of the tangent to the circle at the point \(D\).

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.

7 A circle with centre \(C ( - 3,2 )\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 12$$

1. Find the \(y\)-coordinates of the points where the circle crosses the \(y\)-axis.
2. Find the radius of the circle.
3. The point \(P ( 2,5 )\) lies outside the circle.
   1. Find the length of \(C P\), giving your answer in the form \(\sqrt { n }\), where \(n\) is an integer.
   2. The point \(Q\) lies on the circle so that \(P Q\) is a tangent to the circle. Find the length of \(P Q\).

7 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0\).

1. Find:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle in the form \(p \sqrt { 2 }\), where \(p\) is an integer.
2. A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
3. A line has equation \(y = 2 k - x\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
   2. Find the values of \(k\) for which the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (c)(ii).

5 A circle with centre \(C ( - 5,6 )\) touches the \(y\)-axis, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-6_444_698_372_680}

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. 1. Verify that the point \(P ( - 2,2 )\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(P\).
   3. The mid-point of \(P C\) is \(M\). Determine whether the point \(P\) is closer to the point \(M\) or to the origin \(O\).

3 A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 14 y + 25 = 0$$

1. Write the equation of \(C\) in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where \(a , b\) and \(k\) are integers.
2. Hence, for the circle \(C\), write down:
   1. the coordinates of its centre;
   2. its radius.
3. 1. Sketch the circle \(C\).
   2. Write down the coordinates of the point on \(C\) that is furthest away from the \(x\)-axis.
4. Given that \(k\) has the same value as in part (a), describe geometrically the transformation which maps the circle with equation \(( x + 1 ) ^ { 2 } + y ^ { 2 } = k\) onto the circle \(C\).

5 A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
4. The point \(T\) lies on the tangent to the circle at \(A\) such that \(A T = 4\). Find the length of \(C T\).

7. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 8 y + k = 0 ,$$ where \(k\) is a constant. Given that the point with coordinates \(( - 6,5 )\) lies on \(C\),

1. find the value of \(k\),
2. find the coordinates of the centre and the radius of \(C\). A straight line which passes through the point \(A ( 2,3 )\) is a tangent to \(C\) at the point \(B\).
3. Find the length \(A B\) in the form \(k \sqrt { 3 }\).

7. The circle \(C\) has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).

1. Find the length of the diameter of \(C\).
2. Find an equation for \(C\).
3. Show that the line \(y = 2 x - 3\) is a tangent to \(C\) and find the coordinates of the point of contact.

8. \begin{figure}[h] \end{figure} Figure 2 shows the circle \(C\) and the straight line \(l\). The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).

1. Find the gradient of \(l\).
2. Find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the centre of \(C\).
4. Show that \(C\) has the equation \(x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0\).

6. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0 .$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\).
3. Sketch C. Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).

1. The point \(A\) has coordinates ( 4,6 ).

Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),

1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
2. Find the coordinates of \(B\).
3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   

9 An ellipse is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-5_453_633_365_699} The ellipse intersects the \(x\)-axis at the points \(A\) and \(B\). The equation of the ellipse is $$\frac { ( x - 4 ) ^ { 2 } } { 4 } + y ^ { 2 } = 1$$

1. Find the \(x\)-coordinates of \(A\) and \(B\).
2. The line \(y = m x ( m > 0 )\) is a tangent to the ellipse, with point of contact \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$\left( 1 + 4 m ^ { 2 } \right) x ^ { 2 } - 8 x + 12 = 0$$
   2. Hence find the exact value of \(m\).
   3. Find the coordinates of \(P\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\)

The point \(P\) on \(C\) has \(y\) coordinate \(p\), where \(p\) is a positive constant.

1. Show that an equation of the tangent to \(C\) at \(P\) is given by $$2 p y = 16 x + p ^ { 2 }$$ $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
2. Write down the equation of the directrix of \(C\). The line \(l\) is the reflection of the tangent to \(C\) at \(P\) in the directrix of \(C\).
   Given that \(l\) passes through the focus of \(C\),
3. determine the exact value of \(p\).