Rectangular Cartesian coordinates：练习题

A line \(L\) passes through the point \((3, -2)\) and is parallel to the line with equation \(4x - 2y = 7\). Find the equation of line \(L\).

Find the equation of the perpendicular bisector of the line segment joining the points \(A(2, -3)\) and \(B(6, 1)\).

The point \(P\) divides the line segment joining \(A(-2, 5)\) and \(B(k, -10)\) internally in the ratio \(3:2\). Given that \(P\) lies on the line with equation \(2x + y = 1\), find the value of \(k\).

The line segment joining the points \(P(-1, 4)\) and \(Q(a, b)\) has a gradient of \(-\frac{1}{2}\) and a length of \(\sqrt{20}\). Find the two possible values of \(a\).

The line \(ax + by + c = 0\) passes through the point \((2, 3)\) and is perpendicular to the line \(4x - 5y + 8 = 0\). Find the ratio \(a:b:c\) using the simplest integer values.

Find the equation of the straight line which passes through the point \((4, -1)\) and is parallel to the line with equation \(3x + 2y = 6\).

The line \(L_1\) has the equation \(2x - 3y = 7\) and the line \(L_2\) has the equation \(x + 4y = -2\).

A parallelogram \(PQRS\) has vertices \(P(1, 1)\), \(Q(5, 3)\), and \(R(2, 6)\).