QUESTION

# By what ratio is the line segment joining (0, 2) and (6,8) divided by the line $2x+2y=5$?

Hint: We will use the concept of internal divisions with section formula to solve this question. We will find the point of intersection between line segment AB and the given line equation $2x+2y=5$ using the section formula $P=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\,\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)$ and then substitute x-coordinate and y-coordinate of the point P in the line equation.

Before proceeding with the question we should know about the concept of internal divisions with the section formula.
If point P(x,y) lies on line segment AB between points A and B and satisfies $\text{AP:PB=m:n}$, then we say that P divides AB internally in the ratio m:n. The point of division has the coordinates
$P=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\,\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right).........(1)$
Let the points of the line segment AB are A(0,2) and B(6,8).
The given line in the question is $2x+2y=5$.
Let the line $2x+2y=5$ divide line segment AB in the ratio $k:1$ at point P.
From equation (1) we get the coordinates of point P $=\left( \dfrac{6k+0}{k+1},\,\dfrac{8k+2}{k+1} \right)$
From the question we can conclude that point P lies on the line $2x+2y=5.......(2)$
So now substituting $x=\dfrac{6k}{k+1}$ and $y=\dfrac{8k+2}{k+1}$ in equation (2) we get,
$\,\Rightarrow 2\left( \dfrac{6k}{k+1} \right)+2\left( \dfrac{8k+2}{k+1} \right)=5.......(3)$
Rearranging and simplifying equation (3) we get,
$\,\Rightarrow \dfrac{12k}{k+1}+\dfrac{16k+4}{k+1}=5.......(4)$
Taking L.C.M in equation (4) we get,
$\,\Rightarrow \dfrac{12k+16k+4}{k+1}=5.......(5)$
Now cross multiplying and shifting similar terms to one side and the numbers to another side and then solving for k we get,
\begin{align} & \,\Rightarrow 28k+4=5k+5 \\ & \,\Rightarrow 28k-5k=5-4 \\ & \,\Rightarrow 23k=1 \\ & \,\Rightarrow k=\dfrac{1}{23} \\ \end{align}
So the ratio will be $\dfrac{1}{23}:1$ which can be written as 1:23.
Hence the ratio in which the line segment AB gets divided by $2x+2y=5$ is $1:23$ internally.

Note: We will have to remember the section formula and the important thing here is to assume the ratio $k:1$ because it will reduce one variable. We can substitute y coordinate in place of x coordinate in the section formula by mistake so we need to be careful while doing the substitution of coordinates in the section formula.