高级数学 | 高一下册 | 第十二章 | 12.6 点到直线的距离 Distance from a Point to Line - 独中课程 | 线上补习

高中一 | 高级数学

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12.6 Distance from a Point to Line 点到直线的距离

(1)
Find the distance from the point (2, 3) to the line 7x – 24y + 8 = 0 and prove that the point(2, 3) and the origin are at the different side of the line.

求点(2, 3)到直线7x – 24y + 8 – 0的距离及证明点(2, 3)与原点在直线的两边.

Answer:

(2)
If P(h, k) is collinear with ( –2, 1), (1, 2), and the distance from P to the line 3x – 4y + 5 = 0 is 3 units, find the coordinate of P.

若点P(h, k)与(-2, 1), (1, 2)共线,及P与直线 3x – 4y + 5 = 0 之距离为3单位, 求P的座标。

Answer:

(10, 5), (-8, -1)

(3)
Find the equation of a straight line L, which is parallel to the line x + 2y –3 = 0 and the distance between them is 2 units.

求和直线x + 2y – 3 = 0 平行且距离等于2的直线L的方程式。

Answer:

x + 2y – 3 $\sqrt{5}$ 2 = 0

(4)
Two parallel lines L $_1$ : 2x + 2y – 1 = 0 and L $_2$ : 2x + 2y – 13 = 0.
二平行线L $_1$ : 2x + 2y – 1 = 0及L $_2$ : 2x + 2y – 13 = 0.

(a) Find the y-intercept of L $_1$ .
求L $_1$ 之y截距.

(b) Find the distance between L $_1$ and L $_2$ .
求L $_1$ 及L $_2$ 之距离.

(c) L $_3$ is another line parallel to L $_1$ . If the distance between L $_1$ and L $_3$ is equal to that between L $_1$ and L $_2$ , find the equation of L $_3$ .
L $_3$ 为L $_1$ 的另一平行线. 若L $_1$ 与L $_3$ 的距离和L $_1$ 与L $_2$ 的距离相等, 求L $_3$ 之方程式.

Answer:

(a) $\frac{1}{2}$

(b) 3 $\sqrt{2}$

(5)
Find the equation of line $l$ which parallel with $l_1$ : 2x – 6y + 1 = 0 and $l_2$ : x – 3y – 1 = 0, and separate $l_1$ and $l_2$
求直线 $l$ 之方程式使其与 $l_1$ : 2x – 6y + 1 = 0 和 $l_2$ : x – 3y – 1 = 0平行, 且与 $l_1$ 和 $l_2$

(a) internally to (內分) 1 : 2,

(b) externally to (外分) 1 : 3

Answer:

(a) x – 3y = 0

(b) 4x – 12y + 5 = 0

(6)
Given that the lines x + y + 3 = 0, x + 3y +7 = 0 and 2x + 3y + 5 = 0 form a triangle; find the heights of the triangle.

已知x + y+ 3 = 0, x + 3y + 7 = 0及2x + 3y + 5 = 0形成一三角形, 求其高

Answer:

$\frac{3}{\sqrt{13}}$ , $\frac{6}{\sqrt{10}}$ , $\sqrt{2}$

(7)
ABCD is a trapezium where AB parallel to DC with the coordinates A(0,5), B(4,17), C(7, t), D(0, 0). Calculate the value of t and the distance of the two parallel lines.

ABCD为一梯形且AB与DC平行而座标A(0, 5), B(4, 17), C(7, t), D(0, 0). 求t之值及两平行线的距离.

Answer:

21 , $\frac{\sqrt{10}}{2}$

(8)
Given that C(-1, 0) is the center of a square, x + 3y – 5 = 0 is the equation of one of its sides, find the equation of the other three sides.

已知C(-1, 0)为一正方形的中心, 一条边的方程式是x + 3y – 5 = 0, 求其他三边的方程式.

Answer:

3x – y – 3 = 0, 3x – y + 9 = 0, x + 3y + 7 = 0