LINE JOINING TWO POINTS IS DIVIDED BY THE ANOTHER LINE

Example :

In what ratio does the line x-y-2 = 0 divide the line segment joining the points A (3, -1) and B (8, 9)?

Solution :

Let the line x-y-2  =  0 is divided by the line segment joining the points A(3, -1) and B(8, 9) be a : 1

Using section formula internally :

=  (lx2+mx1)/(l+m), (ly2+my1)/(l+m)

=  a(8)+1(3)/(a+1), a(9)+1(-1)/(a+1)

=  (8a+3)/(a+1), (9a-1)/(a+1)

The point (8a+3)/(a+1), (9a-1)/(a+1) lies on the line.

[(8a+3)/(a+1)]-[(9a-1)/(a+1)]-2  =  0

(8a+3)-(9a-1)-2(a+1)  =  0

8a-9a+3+1-2a-2  =  0

-3a+2  =  0

3a  =  2

a  =  2/3

a : 1  ==>  (2/3) : 1

So, the required ratio is 2 : 3.

Another method :

Equation of the line joining points A (3, -1) and B (8, 9)

(y-y1)/(y2-y1)  =  (x-x1)/(x2-x1)

(y+1)/(9+1)  =  (x-3)/(8-3)

(y+1)/10  =  (x-3)/5

(y+1)/2  =  (x-3)

y+1  =  2(x-3)

y+1  =  2x-6

2x-y-6-1  =  0

2x-y-7  =  0

Point of intersection of the lines 2x-y-7 = 0 and x-y-2 = 0

2x-y-7 = 0 -------(1)

x-y-2 = 0 -------(2)

(1)-(2)

2x-x-7+2  =  0

x-5  =  0

x  =  5

By applying the value of x in (2), we get

5-y-2  =  0

3-y  =  0

y  =  3

So, both are intersecting at the point (5, 3).

Now, we find in what ratio the line joining two points A (3, -1) and B (8, 9) is being divided by the point (5, 3).

(8l+3m)/(l+m), (9l-1m)/(l+m)  =  (5, 3)

Equating x coordinates, we get

(8l+3m)/(l+m)  =  5

8l+3m  =  5l+5m

3l  =  2m

l/m  =  2/3

l:m  =  2 : 3

So, the required ratio is 2 : 3.

Note : By equating y-coordinate also, we will get the same answer.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Problems on Finding Derivative of a Function

    Mar 29, 24 12:11 AM

    Problems on Finding Derivative of a Function

    Read More

  2. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  3. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More