Tangent equation at a point

12 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).

1. Show that the centre of this circle is \(\mathrm { C } ( 4,2 )\) and find the radius of the circle.
2. Show that the origin lies inside the circle.
3. Show that AB is a diameter of the circle, where A has coordinates (2, 7) and B has coordinates \(( 6 , - 3 )\).
4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).

12 You are given that the equation of the circle shown in Fig. 12 is $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$$ \begin{figure}[h] \end{figure}

1. Show that the centre, Q , of the circle is \(( 2,3 )\) and find the radius.
2. The circle crosses the \(x\)-axis at B and C . Show that the coordinates of C are \(( 6,0 )\) and find the coordinates of B .
3. Find the gradient of the line QC and hence find the equation of the tangent to the circle at C.
4. Given that M is the mid-point of BC , find the coordinates of the point where QM meets the tangent at C .

9. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).

1. Find an equation for \(C\).
2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).

1. Show that the centre of this circle is \(C ( 4,2 )\) and find the radius of the circle.
2. Show that the origin lies inside the circle.
3. Show that AB is a diameter of the circle, where A has coordinates ( 2,7 ) and B has coordinates \(( 6 , - 3 )\).
4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).

3 \begin{figure}[h] \end{figure} Not to scale A circle has centre \(\mathrm { C } ( 1,3 )\) and passes through the point \(\mathrm { A } ( 3,7 )\) as shown in Fig. 11.

1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.

10 A circle has centre \(C ( - 2,4 )\) and radius 5 .

1. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. Show that the tangent to the circle at the point \(P ( - 5,8 )\) has equation \(3 x - 4 y + 47 = 0\).
3. Verify that the point \(T ( 3,14 )\) lies on this tangent.
4. Find the area of the triangle \(C P T\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 10 y - 19 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle.
2. Verify that the point \(( 7 , - 2 )\) lies on the circumference of the circle.
3. Find the equation of the tangent to the circle at the point \(( 7 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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

1. Find the coordinates of \(C\) and the radius of the circle.
2. Show that the tangent to the circle at the point \(P ( 8,2 )\) has equation \(3 x + 4 y = 32\).
3. The circle meets the \(y\)-axis at \(Q\) and the tangent meets the \(y\)-axis at \(R\). Find the area of triangle \(P Q R\).

10
\includegraphics[max width=\textwidth, alt={}, center]{0ae3af7e-32cc-43fa-89bb-d6697a8f8061-3_755_905_248_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 6 y - 20 = 0\).

1. Find the centre and radius of the circle. The circle crosses the positive \(x\)-axis at the point \(A\).
2. Find the equation of the tangent to the circle at \(A\).
3. A second tangent to the circle is parallel to the tangent at \(A\). Find the equation of this second tangent.
4. Another circle has centre at the origin \(O\) and radius \(r\). This circle lies wholly inside the first circle. Find the set of possible values of \(r\).

13 \begin{figure}[h] \end{figure} Fig. 13 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 16\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form.
3. Show that the point \(\mathrm { A } ( 4 + 2 \sqrt { 2 } , 2 + 2 \sqrt { 2 } )\) lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it.
   Find the equations of both these tangents.

1. (i) A circle \(C _ { 1 }\) has equation

$$x ^ { 2 } + y ^ { 2 } + 18 x - 2 y + 30 = 0$$ The line \(l\) is the tangent to \(C _ { 1 }\) at the point \(P ( - 5,7 )\).
Find an equation of \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
(ii) A different circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 12 y + k = 0$$ where \(k\) is a constant.
Given that \(C _ { 2 }\) lies entirely in the 4th quadrant, find the range of possible values for \(k\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the circle \(C\)

• the point \(P ( - 1 , k + 8 )\) is the centre of \(C\)
• the point \(Q \left( 3 , k ^ { 2 } - 2 k \right)\) lies on \(C\)
• \(k\) is a positive constant
• the line \(l\) is the tangent to \(C\) at \(Q\)

Given that the gradient of \(l\) is - 2

1. show that $$k ^ { 2 } - 3 k - 10 = 0$$
2. Hence find an equation for \(C\)

3. A circle \(C\) has equation $$x ^ { 2 } - 22 x + y ^ { 2 } + 10 y + 46 = 0$$ a. Find
i. the coordinates of the centre \(A\) of the circle
ii. the radius of the circle. Given that the points \(Q ( 5,3 )\) and \(S\) lie on \(C\) such that the distance \(Q S\) is greatest,
b. find an equation of tangent to \(C\) at \(S\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are constants to be found.

8 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 25\).

1. Write down
   • the radius of the circle,
   • the coordinates of the centre of the circle.
   • Find, in exact form, the coordinates of the points of intersection of the circle with the \(y\)-axis.
   • Show that the point \(( 1,2 )\) lies outside the circle.
   • The point \(\mathrm { P } ( - 1,1 )\) lies on the circle. Find the equation of the tangent to the circle at P .

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 \(D\).

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 \(P Q\).

3 A circle has centre \(C ( 2 , - 1 )\) and radius 5 . The point \(P\) has coordinates \(( 6,2 )\).

1. Write down an equation of the circle.
2. Verify that the point \(P\) lies on the circle.
3. Find the gradient of the line \(C P\).
4. 1. Find the gradient of a line which is perpendicular to \(C P\).
   2. Hence find an equation for the tangent to the circle at the point \(P\).

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 \(k \sqrt { 3 }\).

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.

1. The circle \(C\) has centre \(A\) and equation

$$x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 15 = 0 .$$

1. Find the coordinates of \(A\) and the radius of \(C\).
2. The point \(P\) has coordinates ( \(4 , - 7\) ) and lies on \(C\). Find the equation of the tangent to \(C\) at \(P\).