Tangent equation at a known point on circle

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 \(k\).

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

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 \(D\).

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 \(P\).

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.

The circle \(C\) has equation \(x^2 + y^2 - 8x - 16y - 209 = 0\).

1. Find the coordinates of the centre of \(C\) and the radius of \(C\). [3]

The point \(P(x, y)\) lies on \(C\).

1. Find, in terms of \(x\) and \(y\), the gradient of the tangent to \(C\) at \(P\). [3]
2. Hence or otherwise, find an equation of the tangent to \(C\) at the point \((21, 8)\). [2]

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

1. Find the coordinates of \(C\) and the radius of the circle. [3]
2. Verify that the point \((7, -2)\) lies on the circumference of the circle. [1]
3. Find the equation of the tangent to the circle at the point \((7, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

\includegraphics{figure_10} Fig. 10 shows a circle with centre C\((2, 1)\) and radius 5.

1. Show that the equation of the circle may be written as $$x^2 + y^2 - 4x - 2y - 20 = 0.$$ [3]
2. Find the coordinates of the points P and Q where the circle cuts the \(y\)-axis. Leave your answers in the form \(a \pm \sqrt{b}\). [3]
3. Verify that the point A\((5, -3)\) lies on the circle. Show that the tangent to the circle at A has equation \(4y = 3x - 27\). [6]

A circle has equation \((x - 5)^2 + (y - 2)^2 = 20\).

1. State the coordinates of the centre and the radius of this circle. [2]
2. State, with a reason, whether or not this circle intersects the \(y\)-axis. [2]
3. Find the equation of the line parallel to the line \(y = 2x\) that passes through the centre of the circle. [2]
4. Show that the line \(y = 2x + 2\) is a tangent to the circle. State the coordinates of the point of contact. [5]

\includegraphics{figure_13} 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]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form. [4]
3. Show that the point 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. [7]