Coordinate geometry：練習問題

Determine the gradient of the straight line with the equation \(3x - 2y + 6 = 0\).

Find the equation of the line which passes through the point \((1, 2)\) and is perpendicular to the line with equation \(y = 2x + 5\).

Find the values of the constant \(k\) for which the line \(y = 2x + k\) is a tangent to the circle \(x^2 + y^2 = 5\).

Find the coordinates of the midpoint of the line segment joining the points A\((2, -3)\) and B\((6, 7)\).

Find the equation of the circle with centre \((3, -4)\) that passes through the point \((7, -1)\).

Find the gradient of a line that is perpendicular to the straight line passing through the points A \( (1, 4) \) and B \( (3, 10) \).

A circle has the equation \( (x - 2)^2 + (y + 3)^2 = 25 \). Find the coordinates of the two points where the circle intersects the \( x \)-axis.

Point A has coordinates \( (1, 2) \) and point B has coordinates \( (5, 10) \). Find the equation of the perpendicular bisector of the line segment AB, giving your answer in the form \( ay + bx = c \).

A straight line passes through the points \( A(-2, 5) \) and \( B(4, 13) \).

(a) Find the equation of the line \( AB \), giving your answer in the form \( y = mx + c \).
(b) Find the length of the line segment \( AB \).

The circle \( C \) has the equation \( (x-3)^2 + (y+2)^2 = 25 \).

(a) Write down the coordinates of the centre of the circle and its radius.
(b) Find the equation of the tangent to the circle at the point \( (6, 2) \), giving your answer in the form \( ax + by + c = 0 \).