Question

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

Answer 1

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.

Complete step-by-step answer:
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,

& \,\Rightarrow 28k+4=5k+5 \\\ & \,\Rightarrow 28k-5k=5-4 \\\ & \,\Rightarrow 23k=1 \\\ & \,\Rightarrow k=\dfrac{1}{23} \\\ \end{aligned}$$ So the ratio will be $\dfrac{1}{23}:1$ which can be written as 1:23. Hence the ratio in which the line segment AB gets divided by $$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.