Coordinate geometry in two dimensions：練習問題

A circle has the equation \(x^2 + y^2 - 4x + 6y - 12 = 0\). Find the coordinates of the center and the length of the radius of the circle.

The line segment joining the points \(A(1, 2)\) and \(B(5, 10)\) is the diameter of a circle. Find the equation of the circle.

The line \( y = x + k \) is a tangent to the circle \( x^2 + y^2 - 2x + 4y - 3 = 0 \). Calculate the possible values of \( k \).

A triangle has vertices at A\( (1, 2) \), B\( (5, 4) \) and C\( (k, 8) \). Given that the area of triangle ABC is 10 square units and \( k > 0 \), find the value of \( k \).

A relationship between \( x \) and \( y \) is given by \( y = Ae^{bx} \), where \( A \) and \( b \) are constants. When \( \ln y \) is plotted against \( x \), a straight line passing through the points \( (0, 2) \) and \( (4, 10) \) is obtained. Determine the values of \( A \) and \( b \).

Find the area of the quadrilateral with vertices at \(A(1, 1)\), \(B(5, 2)\), \(C(4, 6)\), and \(D(0, 4)\).

Find the coordinates of the midpoint of the line segment joining \(A(-2, 5)\) and \(B(6, -1)\), and hence determine the equation of the line passing through this midpoint that is perpendicular to the line \(y = \frac{1}{2}x + 4\).

Find the possible values of the constant \(k\) for which the line \(y = 2x + k\) is a tangent to the circle \(x^2 + y^2 - 4x - 2y + 1 = 0\).

A circle has equation \(x^2 + y^2 - 4x + 6y - 12 = 0\).
(a) Find the coordinates of the center and the radius of the circle.
(b) Determine whether the point \(P(5, 1)\) lies inside, outside, or on the circle.

The vertices of a triangle are \(A(-1, 3)\), \(B(5, 1)\), and \(C(3, 7)\).
(a) Find the gradient of \(AB\).
(b) Find the equation of the line passing through \(C\) that is perpendicular to \(AB\).
(c) Calculate the area of triangle \(ABC\).