Tangent equation involving finding the point of tangency

13. The circle \(C\) has centre \(A ( 1 , - 3 )\) and passes through the point \(P ( 8 , - 2 )\).

1. Find an equation for the circle \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(P\).
2. Find an equation for \(l _ { 1 }\), giving your answer in the form \(y = m x + c\) The line \(l _ { 2 }\), with equation \(y = x + 6\), is the tangent to \(C\) at the point \(Q\).
3. Find the coordinates of the point \(Q\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a circle \(C\) with centre \(N ( 4 , - 1 )\). The line \(l\) with equation \(y = \frac { 1 } { 2 } x\) is a tangent to \(C\) at the point \(P\). Find

1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
2. the equation of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{bfeb1724-9a00-4a36-9606-520395792b2b-16_2256_52_311_1978}

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of

• the circle \(C\) with centre \(X ( 4 , - 3 )\)
• the line \(l\) with equation \(y = \frac { 5 } { 2 } x - \frac { 55 } { 2 }\)

Given that \(l\) is the tangent to \(C\) at the point \(N\),

1. show that an equation for the straight line passing through \(X\) and \(N\) is $$2 x + 5 y + 7 = 0$$
2. Hence find
   1. the coordinates of \(N\),
   2. an equation for \(C\).

8. The circle \(C\), with centre at the point \(A\), has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0\). Find

1. the coordinates of \(A\),
2. the radius of \(C\),
3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. Given that the line \(l\) with gradient \(\frac { 7 } { 2 }\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),
4. find an equation of the line which passes through \(A\) and \(T\).

7. \begin{figure}[h] \end{figure} The line \(y = 3 x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(P ( 2,2 )\), as shown in Figure 1. The point \(Q\) is the centre of \(C\).

1. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
2. show that the \(x\)-coordinate of \(Q\) is 5,
3. find an equation for \(C\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a circle \(C\) with centre \(Q\) and radius 4 and the point \(T\) which lies on \(C\). The tangent to \(C\) at the point \(T\) passes through the origin \(O\) and \(O T = 6 \sqrt { } 5\) Given that the coordinates of \(Q\) are \(( 11 , k )\), where \(k\) is a positive constant, (a) find the exact value of \(k\),
(b) find an equation for \(C\).

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

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

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

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 \(C\)

15. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of a circle \(C\) with centre \(N ( 7,4 )\) The line \(l\) with equation \(y = \frac { 1 } { 3 } x\) is a tangent to \(C\) at the point \(P\).
Find

1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
2. an equation for \(C\). The line with equation \(y = \frac { 1 } { 3 } x + k\), where \(k\) is a non-zero constant, is also a tangent to \(C\).
3. Find the value of \(k\).

6. \begin{figure}[h] \end{figure} The circle \(C\) has centre \(A\) with coordinates \(( - 3,1 )\).
The line \(l _ { 1 }\) with equation \(y = - 4 x + 6\), is the tangent to \(C\) at the point \(Q\), as shown in Figure 3.
a. Find the equation of the line \(A Q\) in the form \(a x + b y = c\).
b. Show that the equation of the circle \(C\) is \(( x + 3 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 17\) The line \(l _ { 2 }\) with equation \(y = - 4 x + k , k \neq 6\), is also a tangent to \(C\).
c. Find the value of the constant \(k\).

6. \begin{figure}[h] \end{figure} Not to scale The circle \(C\) has centre \(A\) with coordinates (7,5).
The line \(l\), with equation \(y = 2 x + 1\), is the tangent to \(C\) at the point \(P\), as shown in Figure 3 .

1. Show that an equation of the line \(P A\) is \(2 y + x = 17\)
2. Find an equation for \(C\). The line with equation \(y = 2 x + k , \quad k \neq 1\) is also a tangent to \(C\).
3. Find the value of the constant \(k\).

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.

4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0\).

1. By completing the square, express this equation in the form $$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle, leaving your answer in surd form.
3. A line has equation \(y = 2 x\).
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation \(x ^ { 2 } - 4 x + 4 = 0\).
   2. Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, \(P\).
4. Prove that the point \(Q ( - 1,4 )\) lies inside the circle.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x - 30 y + 209 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). The line \(L\) has equation \(y = m x + 1\), where \(m\) is a constant.
      Given that \(L\) is the tangent to \(C\) at the point \(P\),
2. show that $$2 m ^ { 2 } - 7 m - 22 = 0$$
3. Hence find the possible pairs of coordinates of \(P\).

The line \(L\) has equation $$5y + 12x = 298$$ A circle, \(C\), has centre \((7, 9)\) \(L\) is a tangent to \(C\).

1. Find the coordinates of the point of intersection of \(L\) and \(C\). Fully justify your answer. [5 marks]
2. Find the equation of \(C\). [3 marks]

In this question you must show detailed reasoning. The lines \(y = \frac{1}{2}x\) and \(y = -\frac{1}{2}x\) are tangents to a circle at \((2, 1)\) and \((-2, 1)\) respectively. Find the equation of the circle in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are constants. [6]

In this question you must show detailed reasoning. A circle touches the lines \(y = \frac{1}{2}x\) and \(y = 2x\) at \((6, 3)\) and \((3, 6)\) respectively. \includegraphics{figure_6} Find the equation of the circle. [7]