Tangent from external point - intersection or geometric properties

Find the coordinates of tangent touch points, intersection of tangents, or prove geometric properties (e.g. angles, chord of contact) involving tangents from an external point.

10 questions · Standard +0.5

1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle
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CAIE P1 2021 June Q10
8 marks Standard +0.3
10 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0\).
  1. Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
  2. Find the point of intersection of the tangents to the circle at \(A\) and \(B\).
CAIE P1 2024 June Q8
8 marks Standard +0.3
8 A circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 2 y - 15 = 0\) meets the \(y\)-axis at the points \(A\) and \(B\). The tangents to the circle at \(A\) and \(B\) meet at the point \(P\). Find the coordinates of \(P\). \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_71_1659_466_244} \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_2723_37_136_2010}
CAIE P1 2020 November Q9
9 marks Standard +0.8
9 A circle has centre at the point \(B ( 5,1 )\). The point \(A ( - 1 , - 2 )\) lies on the circle.
  1. Find the equation of the circle.
    Point \(C\) is such that \(A C\) is a diameter of the circle. Point \(D\) has coordinates (5, 16).
  2. Show that \(D C\) is a tangent to the circle.
    The other tangent from \(D\) to the circle touches the circle at \(E\).
  3. Find the coordinates of \(E\).
CAIE P1 2020 November Q11
12 marks Standard +0.8
11 A circle with centre \(C\) has equation \(( x - 8 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100\).
  1. Show that the point \(T ( - 6,6 )\) is outside the circle.
    Two tangents from \(T\) to the circle are drawn.
  2. Show that the angle between one of the tangents and \(C T\) is exactly \(45 ^ { \circ }\).
    The two tangents touch the circle at \(A\) and \(B\).
  3. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
  4. Find the \(x\)-coordinates of \(A\) and \(B\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
Edexcel C2 2014 January Q8
11 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e7043e7a-2c8f-425a-8471-f647828cc297-22_1015_1542_267_185} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a circle \(C\) with centre \(O\) and radius 5
  1. Write down the cartesian equation of \(C\). The points \(P ( - 3 , - 4 )\) and \(Q ( 3 , - 4 )\) lie on \(C\).
  2. Show that the tangent to \(C\) at the point \(Q\) has equation $$3 x - 4 y = 25$$
  3. Show that, to 3 decimal places, angle \(P O Q\) is 1.287 radians. The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) intersect on the \(y\)-axis at the point \(R\).
  4. Find the area of the shaded region \(P Q R\) shown in Figure 2. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-25_177_154_2576_1804}
Edexcel AEA 2006 June Q4
14 marks Challenging +1.2
4.The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$
  1. Show that \(m\) satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin \(O\) to \(C _ { 1 }\) touch \(C _ { 1 }\) at the points \(A\) and \(B\) .
  2. Find the coordinates of the points \(A\) and \(B\) .
    (8)
    Another circle \(C _ { 2 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 13\) .The tangents from the point \(( 4 , - 7 )\) to \(C _ { 2 }\) touch it at the points \(P\) and \(Q\) .
  3. Find the coordinates of either the point \(P\) or the point \(Q\) .
    (2)
OCR H240/01 2020 November Q11
10 marks Challenging +1.2
    1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 10 ( m + 1 ) x + 40 = 0\).
    2. Hence determine the equation of the tangent to the circle at \(A\) which passes through \(P\). [4] A second tangent is drawn from \(P\) to meet the circle at a second point \(B\). The equation of this tangent is of the form \(y = n x + 2\), where \(n\) is a constant less than 1 .
  2. Determine the exact value of \(\tan A P B\).
CAIE P1 2024 June Q10
8 marks Standard +0.8
The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]
Edexcel AS Paper 1 Specimen Q17
10 marks Standard +0.3
A circle \(C\) with centre at \((-2, 6)\) passes through the point \((10, 11)\).
  1. Show that the circle \(C\) also passes through the point \((10, 1)\). [3]
The tangent to the circle \(C\) at the point \((10, 11)\) meets the \(y\) axis at the point \(P\) and the tangent to the circle \(C\) at the point \((10, 1)\) meets the \(y\) axis at the point \(Q\).
  1. Show that the distance \(PQ\) is \(58\) explaining your method clearly. [7]
WJEC Unit 1 Specimen Q8
6 marks Moderate -0.3
The circle \(C\) has radius 5 and its centre is the origin. The point \(T\) has coordinates \((11, 0)\). The tangents from \(T\) to the circle \(C\) touch \(C\) at the points \(R\) and \(S\).
  1. Write down the geometrical name for the quadrilateral \(ORTS\). [1]
  2. Find the exact value of the area of the quadrilateral \(ORTS\). Give your answer in its simplest form. [5]