1.03e Complete the square: find centre and radius of circle

280 questions

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SPS SPS FM 2025 October Q7
7 marks Standard +0.8
The circle \(x^2 + y^2 + 2x - 14y + 25 = 0\) has its centre at the point C. The line \(7y = x + 25\) intersects the circle at points A and B. Prove that triangle ABC is a right-angled triangle. [7]
SPS SPS SM 2025 October Q13
9 marks Standard +0.3
The circle \(C\) has equation $$x^2 + y^2 + 10x - 4y + 1 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\)
    2. the exact radius of \(C\) [2]
    The line with equation \(y = k\), where \(k\) is a constant, cuts \(C\) at two distinct points.
  2. Find the range of values for \(k\), giving your answer in set notation. [2]
  3. The line with equation \(y = mx + 4\) is a tangent to \(C\). Find possible exact values of \(m\). [5]
SPS SPS SM 2025 November Q8
11 marks Standard +0.3
The circles \(C_1\) and \(C_2\) have respective equations $$x^2 + y^2 - 6x - 2y = 15$$ $$x^2 + y^2 - 18x + 14y = 95.$$
  1. By considering the coordinates of the centres and the lengths of the radii of \(C_1\) and \(C_2\), show that \(C_1\) and \(C_2\) touch internally at some point \(P\). [4]
  2. Determine the coordinates of \(P\). [3]
  3. Find the equation of the common tangent to the circles at P. [4]
OCR H240/03 2018 December Q3
5 marks Easy -1.8
\includegraphics{figure_3} The diagram shows a circle with centre \((a, -a)\) that passes through the origin.
  1. Write down an equation for the circle in terms of \(a\). [2]
  2. Given that the point \((1, -5)\) lies on the circle, find the exact area of the circle. [3]
Pre-U Pre-U 9794/2 2010 June Q8
14 marks Standard +0.8
The point \(F\) has coordinates \((0, a)\) and the straight line \(D\) has equation \(y = b\), where \(a\) and \(b\) are constants with \(a > b\). The curve \(C\) consists of points equidistant from \(F\) and \(D\).
  1. Show that the cartesian equation of \(C\) can be expressed in the form $$y = \frac{1}{2(a-b)}x^2 + \frac{1}{2}(a+b).$$ [3]
  2. State the \(y\)-coordinate of the lowest point of the curve and prove that \(F\) and \(D\) are on opposite sides of \(C\). [2]
    1. The point \(P\) on the curve has \(x\)-coordinate \(\sqrt{a^2 - b^2}\), where \(|a| > |b|\). Show that the tangent at \(P\) passes through the origin. [4]
    2. The tangent at \(P\) intersects the line \(D\) at the point \(Q\). In the case that \(a = 12\) and \(b = -8\), find the coordinates of \(P\) and \(Q\). Show that the length of \(PQ\) can be expressed as \(p\sqrt{q}\), where \(p = 2q\). [5]