Coordinate geometry：練習問題

The line \( L \) has equation \( ax + 5y - 8 = 0 \). Given that \( L \) is perpendicular to the line \( 2x - y + 7 = 0 \), find the value of the constant \( a \).

The line \( y = x \) and the curve \( y = x^2 - 6 \) intersect at two points. Which of the following is one of these points of intersection?

The parametric equations of a curve are \( x = 3\cos \theta \) and \( y = 4\sin \theta \). A line \( L \) passes through the origin and the point on the curve where \( \theta = \frac{\pi}{4} \). Find the gradient of line \( L \).

Find the coordinates of the midpoint of the line segment joining the points \( P(-2, 5) \) and \( Q(6, -1) \).

A circle has a diameter with endpoints at \( A(1, 2) \) and \( B(7, 10) \). Find the equation of this circle.

Find the coordinates of the midpoint of the line segment joining the points \(A(2, 8)\) and \(B(10, -4)\).

The circle \( C \) has equation \( x^2 + y^2 + 6x - 8y = 0 \). Find the exact radius of the circle and the coordinates of its centre.

The line \( L \) has equation \( y = mx - 2 \) and the curve \( C \) has equation \( y = x^2 + 6x + 7 \). Find the range of values of \( m \) for which the line and curve do not intersect.

A straight line \( L \) passes through the points \( A(-3, 1) \) and \( B(5, 7) \).
(a) Calculate the exact length of the line segment \( AB \).
(b) Find the equation of the line that is perpendicular to \( L \) and passes through the midpoint of \( AB \). Give your answer in the form \( ax + by + c = 0 \), where \( a, b, \) and \( c \) are integers.

The circle \( C \) has the equation \( x^2 + y^2 - 10x + 4y + 9 = 0 \).
(a) Find the coordinates of the centre and the radius of \( C \).
(b) The line \( L \) has the equation \( y = x - 3 \). Find the coordinates of the points \( A \) and \( B \) where \( L \) intersects \( C \).
(c) Calculate the exact length of the chord \( AB \).