Tangent equation at a known point on circle

7 The point \(A\) has coordinates \(( 1,5 )\) and the line \(l\) has gradient \(- \frac { 2 } { 3 }\) and passes through \(A\). A circle has centre \(( 5,11 )\) and radius \(\sqrt { 52 }\).

1. Show that \(l\) is the tangent to the circle at \(A\).
2. Find the equation of the other circle of radius \(\sqrt { 52 }\) for which \(l\) is also the tangent at \(A\).

8 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + a x + b y - 12 = 0\). The points \(A ( 1,1 )\) and \(B ( 2 , - 6 )\) lie on the circle.

1. Find the values of \(a\) and \(b\) and hence find the coordinates of the centre of the circle.
2. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y = k\), where \(p , q\) and \(k\) are integers.

9 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-12_583_661_260_740} The diagram shows a circle with centre \(A\) passing through the point \(B\). A second circle has centre \(B\) and passes through \(A\). The tangent at \(B\) to the first circle intersects the second circle at \(C\) and \(D\). The coordinates of \(A\) are ( \(- 1,4\) ) and the coordinates of \(B\) are ( 3,2 ).

1. Find the equation of the tangent CBD.
2. Find an equation of the circle with centre \(B\).
3. Find, by calculation, the \(x\)-coordinates of \(C\) and \(D\). \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-14_746_652_262_744} The diagram shows a sector \(C A B\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D , E\) and \(F\), where \(C O D\) is a straight line and angle \(A C D\) is \(\theta\) radians.

9. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 9 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\). The point \(P ( - 2,7 )\) lies on \(C\).
3. Find an equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   

15. \begin{figure}[h] \end{figure} Diagram not drawn to scale The circle shown in Figure 4 has centre \(P ( 5,6 )\) and passes through the point \(A ( 12,7 )\). Find

1. the exact radius of the circle,
2. an equation of the circle,
3. an equation of the tangent to the circle at the point \(A\). The circle also passes through the points \(B ( 0,1 )\) and \(C ( 4,13 )\).
4. Use the cosine rule on triangle \(A B C\) to find the size of the angle \(B C A\), giving your answer in degrees to 3 significant figures.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x + p y + 123 = 0$$ where \(p\) is a constant. Given that the point \(( 1,16 )\) lies on \(C\),

1. find
   1. the value of \(p\),
   2. the coordinates of the centre of \(C\),
   3. the radius of \(C\).
2. Find an equation of the tangent to \(C\) at the point ( 1,16 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-31_33_19_2668_1896}

14. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 16 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\). Given that the radius of \(C\) is 10
2. find the value of \(k\). The point \(A ( a , - 16 )\), where \(a > 0\), lies on the circle \(C\). The tangent to \(C\) at the point \(A\) crosses the \(x\)-axis at the point \(D\) and crosses the \(y\)-axis at the point \(E\).
3. Find the exact area of triangle \(O D E\).

9. A circle \(C\) has equation $$( x - k ) ^ { 2 } + ( y - 2 k ) ^ { 2 } = k + 7$$ where \(k\) is a positive constant.

1. Write down, in terms of \(k\),
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\). Given that the point \(P ( 2,3 )\) lies on \(C\)
2. 1. show that \(5 k ^ { 2 } - 17 k + 6 = 0\)
   2. hence find the possible values of \(k\). The tangent to the circle at \(P\) intersects the \(x\)-axis at point \(T\).
      Given that \(k < 2\)
3. calculate the exact area of triangle \(O P T\).

2. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 10 y - 21 = 0$$ Find

1. 1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). The point \(P ( 5,4 )\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

1. A circle \(C\) has centre \(( 2,5 )\)

Given that the point \(P ( 8 , - 3 )\) lies on \(C\)

1. 1. find the radius of \(C\)
   2. find an equation for \(C\)
2. Find the equation of the tangent to \(C\) at \(P\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 3 shows

• the curve \(C _ { 1 }\) with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 3 x + 14\)
• the circle \(C _ { 2 }\) with centre \(T\)

The point \(T\) is the minimum turning point of \(C _ { 1 }\) Using Figure 3 and calculus,

1. find the coordinates of \(T\) The curve \(C _ { 1 }\) intersects the circle \(C _ { 2 }\) at the point \(A\) with \(x\) coordinate 2
2. Find an equation of the circle \(C _ { 2 }\) The line \(l\) shown in Figure 3, is the tangent to circle \(C _ { 2 }\) at \(A\)
3. Show that an equation of \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 22 } { 3 }$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C _ { 1 } , l\) and the \(y\)-axis.
4. Find the exact area of \(R\).

5. The circle \(C\) has centre \(( 3,1 )\) and passes through the point \(P ( 8,3 )\).

1. Find an equation for \(C\).
2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2. A circle \(C\) with centre at the point \(( 2 , - 1 )\) passes through the point \(A\) at \(( 4 , - 5 )\).

1. Find an equation for the circle \(C\).
2. Find an equation of the tangent to the circle \(C\) at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

3. \begin{figure}[h] \end{figure} Diagram not drawn to scale The circle \(C\) has centre \(P ( 7,8 )\) and passes through the point \(Q ( 10,13 )\), as shown in Figure 2.

1. Find the length \(P Q\), giving your answer as an exact value.
2. Hence write down an equation for \(C\). The line \(l\) is a tangent to \(C\) at the point \(Q\), as shown in Figure 2.
3. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 2 x + 14 y = 0$$ Find

1. the coordinates of the centre of \(C\),
2. the exact value of the radius of \(C\),
3. the \(y\) coordinates of the points where the circle \(C\) crosses the \(y\)-axis.
4. Find an equation of the tangent to \(C\) at the point ( 2,0 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

9

1. Find the equation of the circle with radius 10 and centre ( 2,1 ), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. The circle passes through the point \(( 5 , k )\) where \(k > 0\). Find the value of \(k\) in the form \(p + \sqrt { q }\).
3. Determine, showing all working, whether the point \(( - 3,9 )\) lies inside or outside the circle.
4. Find an equation of the tangent to the circle at the point ( 8,9 ).

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 \(y = m x + c\).

11 \begin{figure}[h] \end{figure} A circle has centre \(C ( 1,3 )\) and passes through the point \(A ( 3,7 )\) 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.

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}

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 \(r\).