Graphs：練習問題

A straight line passes through the points \((2, -5)\) and \((6, 7)\). Work out the gradient of this line.

The equation of the line \(L_1\) is \(y = 2x + 5\).
Find the equation of the line \(L_2\) which is perpendicular to \(L_1\) and passes through the point \((4, 1)\).

A line segment has endpoints at coordinates \(A(-4, 7)\) and \(B(2, -3)\). Determine the coordinates of the midpoint of AB and find the gradient of the line passing through these two points.

A curve has the equation \(y = x^2 - 4x + 1\).
By drawing a suitable straight line on the same grid as the curve, the solutions to the equation \(x^2 - 6x - 2 = 0\) can be found.
Find the equation of this straight line.

Find the coordinates of the midpoint of the line segment joining the points A\((-3, 8)\) and B\((7, -2)\).

A straight line passes through the point \((0, -4)\) and is perpendicular to the line with equation \(y = \frac{1}{3}x + 7\).
Find the equation of this line in the form \(y = mx + c\).

The line \(L\) passes through the points \(A(2, 5)\) and \(B(6, k)\). The gradient of the line perpendicular to \(L\) is \(-\frac{1}{2}\).
Find the value of \(k\).

A curve has the equation \(y = x^2 - 4x + 7\).
By drawing a suitable straight line on the same axes as the curve, the solutions to the equation \(x^2 - 3x + 1 = 0\) can be found.
Find the equation of this straight line in the form \(y = mx + c\).

Point A has coordinates \((2, -3)\) and point B has coordinates \((6, 5)\).
(a) Find the coordinates of the midpoint of the line segment AB.
(b) Calculate the gradient of the line \(L\) that passes through points A and B.
(c) Find the equation of the line \(L\) in the form \(y = mx + c\).

The point \(P\) has coordinates \( (2, 5) \) and the point \(Q\) has coordinates \( (6, -3) \).
(a) Find the equation of the line \(L_1\) that passes through \(P\) and \(Q\). Give your answer in the form \(y = mx + c\).
(b) The line \(L_2\) is perpendicular to \(L_1\) and passes through the midpoint of the line segment \(PQ\). Find the equation of \(L_2\).
(c) Show that the point \(R(1, -2)\) does not lie on the line \(L_2\).