(x, y) 平面的坐標幾何：练习题

A circle has a diameter with endpoints at \( A(-2, 5) \) and \( B(4, -3) \). Which of the following is the correct equation for this circle?

Find the equation of the line passing through \( (2, -1) \) with a gradient of \( 3 \), giving your answer in the form \( y = mx + c \).

Find the equation of the tangent to the circle \(x^2 + y^2 = 10\) at the point \((1, 3))\).

Find the equation of the line that passes through the point \((4, 1))\) and is parallel to the line \(y = 3x - 5))\).

The line \(y = x + k\) is a tangent to the circle \(x^2 + y^2 = 8\). Find the value(s) of \(k\).

Find the coordinates of the centre and the radius of the circle with the equation \( x^2 + y^2 - 8x + 10y + 5 = 0 \).

A circle has equation \( (x-4)^2 + (y+1)^2 = 25 \). Find the equations of the two tangents to the circle that are parallel to the line \( 3x - 4y + 2 = 0 \).
[6 marks]

A circle passes through the origin and its centre lies on the line \( y = 2x \). Given that the radius of the circle is \( \sqrt{5} \), find the equations of the two possible circles.

The line \(L_1\) has equation \(3x - 4y + 8 = 0\).

(a) Find the gradient of \(L_1\).

(b) A second line \(L_2\) is perpendicular to \(L_1\) and passes through the point \((6, 2)\). Find the equation of \(L_2\) in the form \(ax + by + c = 0\), where \(a, b, c\) are integers.

(c) Find the coordinates of the point of intersection of \(L_1\) and \(L_2\).