Equations of straight lines ：練習問題

A straight line L has a slope of \(\frac{1}{2}\) and passes through the point \(P(4, 1)\). Find the equation of the straight line L, expressed in the general form \(Ax + By + C = 0\).

The inclination of a straight line \(L\) is \(150^\circ\). If \(L\) passes through the point \((\sqrt{3}, 2)\), find the \(x\)-intercept of \(L\).

A straight line L passes through the point P(4, 3) and intersects the positive x-axis and the positive y-axis at points A and B respectively. If O is the origin, find the minimum possible area of riangle OAB.

A straight line L has a slope of 3 and its y-intercept is 4. Find the equation of the line L, expressed in the general form \(Ax + By + C = 0\).

A straight line L passes through the intersection point of the lines \(3x + y = 7\) and \(x - y = 1\). If L is perpendicular to the line \(x - 2y + 5 = 0\), find the equation of L.

A straight line L is parallel to the line \(4x + 3y - 12 = 0\). If the distance from the origin \((0, 0)\) to line L is \(2\) units, find the possible equations of L.

A straight line L passes through the point $P(1, 4)$ and intersects the positive $x$-axis at point $A$ and the positive $y$-axis at point $B$. If the area of the triangle formed by L and the coordinate axes is 8 square units, find the equation of the straight line L.

A straight line L has positive x-intercept and y-intercept such that the x-intercept is three times the y-intercept. If L passes through the point \((6, 2)\), find the equation of L.

(a) A straight line \(L\) passes through the points \(P(2, 5)\) and \(Q(6, 1)\). Find the equation of line \(L\). Express your answer in the general form \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are integers.

(b) Determine the coordinates of the \(x\)-intercept and the \(y\)-intercept of the line \(L\).

Consider a straight line \(L_1\) that passes through the points \(A(-1, 4)\) and \(B(5, 1)\).

(a) Find the equation of \(L_1\). Express your answer in the general form \(Ax + By + C = 0\).

(b) Another straight line \(L_2\) is perpendicular to \(L_1\) and passes through the point \(C(3, -2)\). Find the equation of \(L_2\) in the general form.

(c) Let \(P\) be the point of intersection of \(L_2\) and the \(x\)-axis. Find the coordinates of \(P\).