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第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

第十三章: 方程组 Simultaneous Equations

第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming

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

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
(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}

(c) 2x + 2y + 11 = 0

(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_1l_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