Equations of circles：練習問題

Consider a circle with equation \(x^2 + y^2 + 8x - 2y + k = 0\), where \(k\) is a constant. If the radius of the circle is 4, find the value of \(k\).

Find the equation of the circle whose diameter has endpoints at \(A(-2, 3)\) and \(B(4, -1)\).

A circle C is tangent to the straight line L: \(2x - y + 1 = 0\) at the point \(P(1, 3)\). If the circle C also passes through the point \(Q(0, 5)\), find the equation of the circle C.

A circle is centered at the point \(C(-4, 3)\). If the circle is tangent to the \(y\)-axis, what is the general equation of the circle?

In the rectangular coordinate plane, the circle \( C \) is tangent to both the \(x\)-axis and the \(y\)-axis. If the centre of \( C \) lies in the second quadrant and the radius is 3, find the equation of \( C \).

A circle is defined by the equation \(x^2 + y^2 + 8x - 2y + 8 = 0\). Find the coordinates of its centre and its radius.

A circle passes through the points \(A(1, 4)\) and \(B(5, 0)\). If the centre of the circle lies on the straight line \(2x + 3y = 1\), find the general equation of the circle.

A circle passes through the points \(A(1, 0)\) and \(B(5, 0)\), and is tangent to the straight line \(L: y = 3\). Find the general equation of the circle.

Consider the circle C with equation \(x^2 + y^2 - 8x - 4y + 10 = 0\). Let \(G\) be the centre of \(C\).

(a) Find the coordinates of the centre \(G\) and the radius \(r\) of the circle \(C\).

(b) The straight line \(L\) has the equation \(3x - 4y + 25 = 0\). Determine the relationship between \(L\) and \(C\). Justify your answer using the distance formula.

(c) A circle \(C'\) passes through the centre \(G\) of \(C\) and is tangent to the line \(L\) at the point \(P(1, 7)\). Find the coordinates of the centre \(K\) of \(C'\).

(d) Hence, find the equation of the circle \(C'\) in the general form \(x^2 + y^2 + Dx + Ey + F = 0\).

A circle C is tangent to the line
\(L_1: y = 0\) at the point \(P(1, 0)\). The circle C is also tangent to the line
\(L_2: 3x - 4y = 0\).

(a) Explain why the centre of the circle C must lie on the line \(x=1\), and express the radius \(r\) in terms of the coordinates of the centre.

(b) Let the centre of the circle C be \((1, k)\). Use the condition that the circle is also tangent to \(L_2\) to show that \(5|k| = |3 - 4k|\).

(c) Find the possible coordinates of the centre and the corresponding radii of the circle C.

(d) Hence, write down the general equations of the two possible circles C.