Question

Find the ratios in which the line 3x+4y-9=0 divide the line segment joining the point (1,3) &(2,7)

Explain Briefly!
Don't give irrelevant answers !​

Answer 1

Solution

Given :-

$\sf \implies \: Equation \: 3x + 4y - 9 = 0$

$\sf \implies Point \: (1,3) \: and \: (2,7)$

Let

$\sf \implies \: Ratio \: = P \ratio \: 1$

Using section formula

$\implies \sf \: p \bigg( \dfrac{x_2m + nx_1}{m + n} , \: \dfrac{y_2m + y_1n}{m + n} \bigg)$

Where

$\sf \implies \: x_1 = 1, y_1 = 3 \\ \sf \implies \: x_2 = 2,y_2 = 7$

$\sf \implies \: m = p \: \: and \: n = 1$

Now put the value in formula

$\sf \implies\: p \bigg( \dfrac{2 \times p + 1 \times1}{p + 1} , \: \dfrac{7 \times p + 3 \times 1}{p + 1} \bigg)$

$\sf \implies\: p \bigg( \dfrac{2p + 1 }{p + 1} , \: \dfrac{7 p + 3 }{p + 1} \bigg)$

Now put the value of x and y in Given equation

$\sf \implies \: 3x + 4y - 9 = 0$

$\sf \implies3 \bigg( \dfrac{2p + 1}{p + 1} \bigg) + 4 \bigg( \dfrac{7p + 3}{p + 1} \bigg) - 9 = 0$

$\sf\implies \: \dfrac{6p + 3}{p + 1} + \dfrac{28p + 12}{p + 1} - 9 = 0$

Taking Lcm

$\sf \implies \: \dfrac{6p + 3 + 28p + 12 - 9p - 9}{p + 1} = 0$

$\sf \implies\: 6p + 3 + 28p + 12 - 9p - 9 = 0$

$\sf \implies \: 34p + 15 - 9p - 9 = 0$

$\sf \implies 25p + 6 = 0$

$\sf \implies \: p = \dfrac{ - 6}{25}$

Answer

The ratio is -6/25 or -6:25

Answer 2

Simple method

Let the equation of the other line be y=mx+c

where m is the slope of the line

m=y2-y1/x2-x1=7-3/2-1=4

equation of the line =>y=4x+c

this line passes through (1,3),so simply put it in the equation to take out c

3=4×1+c

c=-1

so, the equation of the line y=4x-1=>4x-y=1

equation of second line =>3x+4y=9

Both line will intersect at the same point

so,equate both of them to take out that point

3x+4y=9....i)

4x-y=1......ii)

multiply second equation by 4

16x-4y=4.....iii)

adding equation i) and iii) we get

19x=13

x=13/19

putting this value in ii) we get

y=71/19

hence the coordinate which divided the line is

(13/19,71/19)

let the ratio in which they divide be k:1

now using section formula we can say

13/19=(mx2+nx1)/(m+n)

now here m=k n=1,x2=2 and x1=1

13/19=(k×2+1×1)/(k+1)

13/19=(2k+1)/(k+1)

13k+13=38k+19

-6=25k

k=-6/25

ration in which the line is divided is -6/25:1 or -6:25

Note:Why ratio is coming negative?

Ans: Negative retion shows that the line is divided externally by the coordinate(13/19,71/19)