Circles: Practice Questions

Find the coordinates of the centre and the radius of the circle with equation \(x^2 + y^2 - 6x + 10y + 9 = 0\).

A circle has equation \(x^2 + y^2 - 8x - 6y + 20 = 0\). Tangents are drawn from the origin \((0,0)\) to the circle.

Find the exact acute angle between these two tangents.

A circle has a diameter with endpoints \(A(-2, 5)\) and \(B(4, -3)\). Find the Cartesian equation of this circle.

The point \(P(5, 7)\) lies on a circle with centre \(C(2, 3)\). Find the equation of the tangent to the circle at point \(P\).

A circle \(C\) has the equation \(x^2 + y^2 - 4x + 6y - 12 = 0\). A point \(P(5, 1)\) lies on the circumference. Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(ax + by + c = 0\).

A circle has the equation \(x^2 + y^2 + 8x - 12y + 27 = 0\).
Find the coordinates of the centre and the radius of the circle.

The circle \(C\) has equation \(x^2 + y^2 - 8x - 6y + 20 = 0\).

(a) Find the coordinates of the centre \(M\) and the radius \(r\) of the circle \(C\).
(b) The line segment joining the points \(A(2, 4)\) and \(B(6, 4)\) is a chord of the circle \(C\).
    (i) Verify that the points \(A\) and \(B\) both lie on \(C\).
    (ii) Find the equation of the perpendicular bisector of the chord \(AB\).
    (iii) Show that this perpendicular bisector passes through the centre \(M\) of the circle.

A circle \(C\) has the equation \(x^2 + y^2 - 10x + 4y + 4 = 0\).
(a) Find the coordinates of the centre and the radius of the circle.
(b) Verify that the point \(P(8, 2)\) lies on the circumference of the circle.
(c) Find the equation of the tangent to the circle at the point \(P\). Give your answer in the form \(y = mx + k\).
(d) A line \(L\) has the equation \(y = kx - 1\). Determine the set of values of the constant \(k\) for which the line \(L\) does not intersect the circle.