Line-circle intersection points

7
\includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-10_887_1003_258_571} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20\) and with centre \(C\). The point \(B\) has coordinates \(( 0,2 )\) and the line segment \(B C\) intersects the circle at \(P\).

1. Find the equation of \(B C\).
2. Hence find the coordinates of \(P\), giving your answer in exact form.

12
\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-18_1006_938_269_591} The diagram shows a circle \(P\) with centre \(( 0,2 )\) and radius 10 and the tangent to the circle at the point \(A\) with coordinates \(( 6,10 )\). It also shows a second circle \(Q\) with centre at the point where this tangent meets the \(y\)-axis and with radius \(\frac { 5 } { 2 } \sqrt { 5 }\).

1. Write down the equation of circle \(P\).
2. Find the equation of the tangent to the circle \(P\) at \(A\).
3. Find the equation of circle \(Q\) and hence verify that the \(y\)-coordinates of both of the points of intersection of the two circles are 11.
4. Find the coordinates of the points of intersection of the tangent and circle \(Q\), giving the answers in surd form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Points \(A ( 7,12 )\) and \(B\) lie on a circle with centre \(( - 2,5 )\). The line \(A B\) has equation \(y = - 2 x + 26\).
Find the coordinates of \(B\).

10 Th circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - \quad\) Ch s cen re \(C\) a¢ sses the id s \(A\) ad \(B\).

1. State th co ida tesg \(C\). It is \(\dot { \mathbf { g } } \dot { \mathrm { n } }\) the t th mid oin, \(D , 6 \quad A B \mathbf { h }\) s co \(\dot { \mathrm { d } } \mathbf { a }\) tes \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
2. Fid eq tin \(A B , \dot { \mathrm {~g} } \dot { \mathrm { v } }\) as wer in th fo \(\mathrm { m } y = m x + c\).
3. Fidy calch atitil b \(x\)-co dia tes \(6 A\) ad \(B\).

13. The point \(A ( 9 , - 13 )\) lies on a circle \(C\) with centre the origin and radius \(r\).

1. Find the exact value of \(r\).
2. Find an equation of the circle \(C\). A straight line through point \(A\) has equation \(2 y + 3 x = k\), where \(k\) is a constant.
3. Find the value of \(k\). This straight line cuts the circle again at the point \(B\).
4. Find the exact coordinates of point \(B\).

1. The circle \(C\)

• has centre \(A ( 3,5 )\)
• passes through the point \(B ( 8 , - 7 )\)
  1. Find an equation for \(C\).

The points \(M\) and \(N\) lie on \(C\) such that \(M N\) is a chord of \(C\).
Given that \(M N\)

• lies above the \(x\)-axis
• is parallel to the \(x\)-axis
• has length \(4 \sqrt { 22 }\)
• find an equation for the line passing through points \(M\) and \(N\).

10 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 8 = 0\).

1. Find the centre and radius of the circle.
2. The circle passes through the point \(( - 3 , k )\), where \(k < 0\). Find the value of \(k\).
3. Find the coordinates of the points where the circle meets the line with equation \(x + y = 6\).

8

1. Describe completely the curve \(x ^ { 2 } + y ^ { 2 } = 25\).
2. Find the coordinates of the points of intersection of the curve \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(2 x + y - 5 = 0\).

9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .

1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
2. Calculate the length of \(A B\), giving your answer in simplified surd form.
3. Find an equation for the line \(A B\).

7. The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).

1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
2. Find the \(x\)-coordinates of \(A\) and \(B\).
3. Show that \(A B = 2 \sqrt { 10 }\).

1 Find the coordinates of the points of intersection of the circle \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(y = 3 x\). Give your answers in surd form.
\(2 \mathrm {~A} ( 9,8 ) , \mathrm { B } ( 5,0 )\) an \(\mathrm { C } ( 3,1 )\) are three points.

1. Show that AB and BC are perpendicular.
2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle.
3. BD is a diameter of the circle. Find the coordinates of D .

5 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).

1. Find the centre and radius of the circle.
2. Find the coordinates of any points where the line \(y = 2 x - 3\) meets the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
3. State what can be deduced from the answer to part (ii) about the line \(y = 2 x - 3\) and the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).

8 In this question you must show detailed reasoning.

1. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 4 y - 20 = 0\).
2. Find the points of intersection of the circle with the line \(x + 3 y - 10 = 0\).

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 \(Q\).

8 A circle has centre \(C ( 3 , - 8 )\) and radius 10 .

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis.
3. The tangent to the circle at the point \(A\) has gradient \(\frac { 5 } { 2 }\). Find an equation of the line \(C A\), giving your answer in the form \(r x + s y + t = 0\), where \(r , s\) and \(t\) are integers.
4. The line with equation \(y = 2 x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.

7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
Find the coordinates of the points of intersection of the line and the circle.

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 \(P Q\).

10 Find the coordinates of the points of intersection of the circle \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(y = 3 x\). Give your answers in surd form.