Coordinate geometry in the (x, y) plane: Practice Questions

The line with equation \(y = x + 7\) intersects the circle with equation \(x^2 + y^2 = 25\) at two points. Calculate the length of the chord formed by these two points.

The circle \(S\) has equation \(x^2 + y^2 - 6x + 8y = 0\). The line \(L\) passes through the center of \(S\) and is perpendicular to the line with equation \(3x - 4y + 5 = 0\). Find the equation of line \(L\).

The circle \(C\) has equation \((x - 3)^2 + (y + 2)^2 = 20\). The point \(P(5, 2)\) lies on the circle. Find the equation of the tangent to the circle at the point \(P\), giving your answer in the form \(ax + by + c = 0\).

The line with equation \(y = kx - 2\) is a tangent to the circle with equation \(x^2 + y^2 - 4x + 2 = 0\). Find the possible values of the constant \(k\).

A circle passes through the points \(A(0, 5)\), \(B(0, -1)\), and \(C(4, -1)\). Find the equation of this circle.

A circle has the equation \((x - 2)^2 + (y + 3)^2 = 25\). Find the equation of the tangent to the circle at the point \((5, 1)\), giving your answer in the form \(ax + by + c = 0\).

Find the equation of the circle where the line segment joining the points A\((-3, 8)\) and B\((5, 2)\) is a diameter.

Show that the line with equation \(y = 2x + 5\) is a tangent to the circle \(x^2 + y^2 = 5\) and find the coordinates of the point of contact.

The circle \(C\) has equation \(x^2 + y^2 - 8x + 6y + 9 = 0\).
(a) Find the coordinates of the centre of \(C\) and the radius of \(C\).
(b) The line \(l\) with equation \(y = x - 1\) intersects the circle \(C\) at points \(A\) and \(B\). Find the exact coordinates of \(A\) and \(B\).