Coordinates：練習問題

A point M is translated from the origin by the vector \(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\). What are the coordinates of point M?

The point \((p, q)\) is the reflection of \((2, 5)\) in the line \(y = 3\). Find the values of \(p\) and \(q\).

Point \(P\) has coordinates \( (3, 4) \). Point \(Q\) is the reflection of point \(P\) in the line \( y = 2x + 1 \). Find the coordinates of \(Q\).

A point is reflected in the \(y\)-axis. If the original coordinates were \((-3, 7)\), what are the coordinates of the reflected point?

A line segment joins point \(M(-1, 4)\) and point \(N(5, 12)\).
Calculate the exact length of the line segment MN.

The line passing through the points \((2, 5)\) and \((6, k)\) is parallel to the line \(y = 2x - 7\). Find the value of \(k\).

A straight line passes through the point \(M(2, 3)\). The midpoint of the line segment between the \(x\)-intercept and the \(y\)-intercept of this line is \(M\). Find the equation of the line.

The point \(C\) lies on the line segment joining \(A(-4, 11)\) and \(B(8, -1)\). If the ratio of the lengths \(AC:CB\) is \(1:3\), find the coordinates of point \(C\).

The line \(L_1\) has the equation \(2y - 6x = 10\).

(a) Find the gradient of line \(L_1\).

(b) Line \(L_2\) is parallel to \(L_1\) and passes through the point \((3, -2)\). Find the equation of line \(L_2\) in the form \(y = mx + c\).