About Lesson
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.
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.
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.
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
2 = 0
![Rendered by QuickLaTeX.com \sqrt{5}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d128f0fae3ed37801eaa4f8f06a376dc_l3.png)
(4)
Two parallel lines L
: 2x + 2y – 1 = 0 and L
: 2x + 2y – 13 = 0.
二平行线L
: 2x + 2y – 1 = 0及L
: 2x + 2y – 13 = 0.
Two parallel lines L
![Rendered by QuickLaTeX.com _1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-3a8732cc72efffee8872fbe29964e4a0_l3.png)
![Rendered by QuickLaTeX.com _2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-5e62d6979a14842f0671d1eb51126a78_l3.png)
二平行线L
![Rendered by QuickLaTeX.com _1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-3a8732cc72efffee8872fbe29964e4a0_l3.png)
![Rendered by QuickLaTeX.com _2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-5e62d6979a14842f0671d1eb51126a78_l3.png)
(a) Find the y-intercept of L.
求L之y截距.
(b) Find the distance between L and L
.
求L及L
之距离.
(c) L is another line parallel to L
. If the distance between L
and L
is equal to that between L
and L
, find the equation of L
.
L为L
的另一平行线. 若L
与L
的距离和L
与L
的距离相等, 求L
之方程式.
Answer:
(a)
![Rendered by QuickLaTeX.com \frac{1}{2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-f5ac0f658c5eea95b9d59561e1d0ecab_l3.png)
(b) 3
(c) 2x + 2y + 11 = 0
(5)
Find the equation of line
which parallel with
: 2x – 6y + 1 = 0 and
: x – 3y – 1 = 0, and separate
and ![Rendered by QuickLaTeX.com l_2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d2923dcfd22c9c6f45082861edd76b54_l3.png)
求直线
之方程式使其与
: 2x – 6y + 1 = 0 和
: x – 3y – 1 = 0平行, 且与
和
Find the equation of line
![Rendered by QuickLaTeX.com l](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-350cd7a522845de2e6287047756ef43b_l3.png)
![Rendered by QuickLaTeX.com l_1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6821630c53f8038dc4d2d65915e6d72c_l3.png)
![Rendered by QuickLaTeX.com l_2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d2923dcfd22c9c6f45082861edd76b54_l3.png)
![Rendered by QuickLaTeX.com l_1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6821630c53f8038dc4d2d65915e6d72c_l3.png)
![Rendered by QuickLaTeX.com l_2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d2923dcfd22c9c6f45082861edd76b54_l3.png)
求直线
![Rendered by QuickLaTeX.com l](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-350cd7a522845de2e6287047756ef43b_l3.png)
![Rendered by QuickLaTeX.com l_1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6821630c53f8038dc4d2d65915e6d72c_l3.png)
![Rendered by QuickLaTeX.com l_2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d2923dcfd22c9c6f45082861edd76b54_l3.png)
![Rendered by QuickLaTeX.com l_1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6821630c53f8038dc4d2d65915e6d72c_l3.png)
![Rendered by QuickLaTeX.com l_2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d2923dcfd22c9c6f45082861edd76b54_l3.png)
(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.
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:
![Rendered by QuickLaTeX.com \frac{3}{\sqrt{13}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-e10c820bb55993a4df1ec84b5a47348f_l3.png)
![Rendered by QuickLaTeX.com \frac{6}{\sqrt{10}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-35d8b910e9fb240f10e5f39e349163f4_l3.png)
![Rendered by QuickLaTeX.com \sqrt{2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6d7301a87ce2408f80e60e536a321415_l3.png)
(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 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 , ![Rendered by QuickLaTeX.com \frac{\sqrt{10}}{2}](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzMyIgaGVpZ2h0PSIzMiIgdmlld0JveD0iMCAwIDMzIDMyIj48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjZWFlYWVjIi8+PC9zdmc+)
![Rendered by QuickLaTeX.com \frac{\sqrt{10}}{2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-6933992c6c880761e424fda364ee13c3_l3.png)
(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.
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