Question

Find the ratio in which the line segment joining the points $\left( 1,2 \right)$ and $\left( -2,3 \right)$ is divided by the line $3x+4y=7$.

Answer 1

Hint: Find the equation of line passing through given points. Then find the intersections point. For line equation through $\left( a,b \right)\left( c,d \right)$ we have formula:
$\left( y-b \right)=\dfrac{\left( d-b \right)}{\left( c-a \right)}\left( x-a \right)$
Use this to find an equation. Substitute the y in terms of x into other equations. Solve for x. Place the value of x back for the value of y. Now take ratio to be $m:1-m$. By formula, if $\left( x,y \right)$ divides $\left( a,b \right);\left( c,d \right)$ by ratio $m:\left( 1-m \right)$ then:
$y=m\left( d \right)+\left( 1-m \right)b$
By using this find m.

Complete step-by-step answer:
Find the equation through given points in the question. Let point is given by the coordinates of the value $A\left( 1,2 \right)$ . Let point B is given by the coordinate of the value $B\left( -2,3 \right)$.
Line equation passing through $\left( a,b \right);\left( c,d \right)$ is given by the formula:
$\left( y-b \right)=\dfrac{\left( d-b \right)}{\left( c-a \right)}\left( x-a \right)$
Here, $a=1,\text{ b=2, c=-2, d=3}$
By substituting this values in the given formula, we get
$\left( y-2 \right)=\dfrac{\left( 3-2 \right)}{\left( -2-1 \right)}\left( x-1 \right)$
So, by cross multiplying the terms the equation, we get:
$-3y+6=x-1$
By adding 3y term on both sides of the equation, we get:
$6=x+3y-1$
By adding 1 on both sides of the equation, we get:
$x+3y=7$
By substituting the value of x in the 2nd line equation. Given in question 2nd line equation is given by equation:
$3x+4y=7$ , we get:
$3\left( 7-3y \right)+4y=7$
By simplifying, we get value of y to be $\dfrac{14}{5}$
By value of y, we get the value of x to be $-\dfrac{7}{5}$ .
Let point $P\left( -\dfrac{7}{5},\dfrac{14}{5} \right)$ divide line in ratio $m:1-m$ .
 By basic knowledge of coordinate geometry, we can say:
$\dfrac{14}{5}=m\times 3+\left( 1-m \right)\cdot 2=3m+2-2m$
By simplifying we get the value of m to be $\dfrac{4}{5}$
Ratio $=m:1-m=\dfrac{4}{5}:\dfrac{1}{5}=4:1$
So, the required ratio is $4:1$

Note: Apply coordinate geometry properly. The required ratio is not m it is $m:1-m$ don’t forget to apply this or else you will get 4:5.
Alternate method:
Instead of using section formulas you can use the distance formula. Find the distance between the intersection point and given 2 points. Then take the ratio of 2 distances to get the required result.
Distance between 2 points (a, b) and (c, d) is given by:
\[\text{Distance}=\sqrt{{{\left( a-c \right)}^{2}}+{{\left( b-d \right)}^{2}}}\]