# Coordinate Geometry

• Dec 15th 2013, 05:57 AM
varunkanpur
Coordinate Geometry
I am not able to solve this problem of Book 'The Element of Co-ordinate of SL Loney'

Find the equations of the two straight lines drawn through
the point (0, a) on which the perpendiculars let fall from the point
(2a, 2a) are each of length a.

• Dec 15th 2013, 06:53 AM
emakarov
Re: Coordinate Geometry
Do you know the formula for the distance from a point to a line, i.e., the length of perpendicular in this case?
• Dec 15th 2013, 03:26 PM
johng
Re: Coordinate Geometry
Hi,
Emakarov has told you how to solve the problem. I recommend that you follow his hint and solve algebraically. However, here's a geometry/trigonometry solution:

Attachment 29919
• Dec 15th 2013, 04:00 PM
romsek
Re: Coordinate Geometry
very elegant
• Dec 15th 2013, 06:39 PM
Soroban
Re: Coordinate Geometry
Hello, varunkanpur!

Quote:

Find the equations of the two straight lines drawn through A(0,a)
on which the perpendiculars from the point B(2a,2a) are each of length a.

Code:

```      |    D       |      *       |    /  * a       | 2a /      *    B       |  /          *(2a,2a)       |  /        *  :       | /    *      :a   A  |/@ * @        :  (0,a)* - - - - - - - * - -       |      2a      :C       |              :       |              :   - - * - - - + - - - + - - -       |      a      2a       |```
We have: ∆ABC: AC = 2a, BC = a.

We see that one line is: .y = a

Reflect ∆ABC over AB so that: ∆ABD ≅ ∆ABC.
Let θ = /BAC = /BAD.

The slope of AB is: .tanθ = 1/2

. . . . . . . . 2tanθ . . . . . .2(1/2) . . . . .1 . . . . 4
tan2θ .= .----------- .= .---------- .= .---- .= .--
. . . . . . . .1 - tan2θ . . . 1 - (1/2)2 . . .3/4 . . . 3

The other line is: .y = (4/3)x + a
• Dec 15th 2013, 06:41 PM
romsek
Re: Coordinate Geometry
why repost what johng has already posted?
• Dec 15th 2013, 11:20 PM
varunkanpur
Re: Coordinate Geometry
Thanx for helping me and taking plains in solving the problem.