Coordinate geometry：练习题

A line passes through the points \(A(2, -1)\) and \(B(8, 7)\). Find the coordinates of the midpoint of the line segment \(AB\).

Points \(A\) and \(B\) have coordinates \((1, 4)\) and \((5, 2)\) respectively. Find the equation of the perpendicular bisector of \(AB\).<\/p>

A circle passes through the points \((0, 0)\), \((8, 0)\) and \((0, 6)\). Find the equation of the circle.

Find the coordinates of the midpoint of the line segment joining the points \(P(-4, 7)\) and \(Q(2, -3)\).<\/p>

The equation of a circle is given by \(x^2 + y^2 - 4x + 6y - 3 = 0\). Find the radius of the circle.

A line segment has endpoints at \(A(1, 4)\) and \(B(7, 10)\). Calculate the coordinates of the midpoint of the line segment \(AB\).

The line \(y = kx - 5\) is a tangent to the curve \(y = x^2 - 4x - 1\). Find the possible values of the constant \(k\).

The line with equation \(3x - 4y + 12 = 0\) intersects the \(x\)-axis at point \(A\) and the \(y\)-axis at point \(B\). Find the equation of the perpendicular bisector of the line segment \(AB\).

The points \(P\) and \(Q\) have coordinates \((1, -2)\) and \((7, 6)\) respectively.
(a) Calculate the length of the line segment \(PQ\).
(b) Find the coordinates of the midpoint of \(PQ\).

The line \(y = x + k\) is a tangent to the curve \(y = x^2 - 5x + 12\).
(a) Show that the \(x\)-coordinate of the point of contact satisfies the equation \(x^2 - 6x + (12 - k) = 0\).
(b) Using the discriminant, find the value of the constant \(k\).