1.03f Circle properties: angles, chords, tangents

103 questions

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SPS SPS SM Pure 2021 June Q11
9 marks Standard +0.3
  1. A circle \(C_1\) has equation $$x^2 + y^2 + 18x - 2y + 30 = 0$$ The line \(l\) is the tangent to \(C_1\) at the point \(P(-5, 7)\). Find an equation of \(l\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found. [5]
  2. A different circle \(C_2\) has equation $$x^2 + y^2 - 8x + 12y + k = 0$$ where \(k\) is a constant. Given that \(C_2\) lies entirely in the 4th quadrant, find the range of possible values for \(k\). [4]
SPS SPS SM Pure 2023 September Q3
7 marks Moderate -0.8
$$x^2 + y^2 - 2x - 2y = 8$$ The circle with the above equation has radius \(r\) and has its centre at the point \(C\).
  1. Determine the value of \(r\) and the coordinates of \(C\). [3]
The point \(P(4,2)\) lies on the circle.
  1. Show that an equation of the tangent to the circle at \(P\) is [4] $$y = 14 - 3x.$$
SPS SPS SM 2023 October Q10
7 marks Standard +0.8
In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]