Coordinate geometry of the circle：練習問題

A circle has its centre at the point \((5, 12)\) and passes through the origin \((0, 0)\). Calculate the radius of the circle.

Determine the points of intersection between the line \(y = 2x\) and the circle \(x^2 + y^2 - 10x = 0\).

A circle has its centre at the point \((2, -3)\) and passes through the point \((5, 1)\). Find the equation of the circle in the form \(x^2 + y^2 + 2gx + 2fy + c = 0\).

Find the equation of the circle with centre \((4, -1)\) and radius \(3\).

The circle \(x^2 + y^2 - 4x + 2y + k = 0\) has a radius of \(3\) units. Find the value of the constant \(k\).

A circle has the equation \(x^2 + y^2 - 8x = 9\). State the coordinates of its centre and the length of its radius.

The line \(y = x + 1\) intersects the circle \(x^2 + y^2 = 13\) at points \(A\) and \(B\). Calculate the length of the chord \(AB\).

A circle passes through the points \(P(0, 5)\) and \(Q(0, -1)\). Given that the centre of the circle lies on the line \(y = x\), find the equation of the circle.

The equation of a circle is given by \(x^2 + y^2 + 6x - 4y - 12 = 0\).
(a) Show that the point \(P(1, 5)\) lies on the circumference of the circle.
(b) Find the coordinates of the point \(Q\) such that the line segment \(PQ\) forms a diameter of the circle.

The points \(A(1, 1)\) and \(B(7, 9)\) are the endpoints of a diameter of circle \(C\).
(a) Find the equation of circle \(C\) in the form \(x^2 + y^2 + gx + fy + c = 0\).
(b) Determine whether the point \(Q(8, 2)\) lies inside, outside, or on the circumference of the circle. Justify your answer.