(x, y) 平面的坐標幾何：練習題

Find the gradient of the straight line passing through the points \((2, -5)\) and \((-4, 7)\).

The circle \(C\) has equation \((x - 5)^2 + (y + 2)^2 = 20\). The line \(L\) is a tangent to the circle at the point \(P(3, 2)\). Find the equation of the line \(L\) in the form \(y = mx + c\).

Find the equation of the straight line that is parallel to the line with equation \( 2x - 5y + 7 = 0 \) and passes through the point \( (10, -3) \). Give your answer in the form \( ax + by + c = 0 \), where \( a, b, \) and \( c \) are integers.

A circle has a diameter with endpoints at the points \( P(-3, 2) \) and \( Q(5, 8) \). Find the equation of the circle in the form \( (x - a)^2 + (y - b)^2 = r^2 \), where \( a, b, \) and \( r \) are constants.

A circle has the equation \( x^2 + y^2 + 8x - 2y - 19 = 0 \).
(a) Find the coordinates of the centre of the circle.
(b) Find the exact value of the radius of the circle.

The circle \( C \) has equation \( x^2 + y^2 - 4x + 6y - 12 = 0 \).
(a) Find the coordinates of the center of \( C \) and the radius of \( C \).
(b) The point \( P(5, 1) \) lies on the circle.
Find the equation of the tangent to the circle at the point \( P \). Give your answer in the form \( ax + by + c = 0 \), where \( a, b \) and \( c \) are integers.

A circle \( C \) has center \( M(4, 2) \) and passes through the point \( A(8, 5) \).
(a) Find an equation for the circle \( C \).
(b) The line \( l \) has equation \( y = x - 3 \). Show that \( l \) is a chord of the circle \( C \) by finding the coordinates of the two points of intersection, \( P \) and \( Q \), between the line and the circle.
(c) Calculate the exact length of the chord \( PQ \).
(d) Find the distance from the center \( M \) to the chord \( PQ \).