Question

Let PS be the median of the triangle with vertices $P\left( 2,2 \right),Q\left( 6,-1 \right)\And R\left( 7,3 \right)$. The equation of line passing through $\left( 1,-1 \right)$ and parallel to PS is:
(a)$4x-7y-11=0$
(b)$2x+9y+7=0$
(c)$4x+7y+3=0$
(d)$2x-9y-11=0$

Answer 1

Hint: We are asked to find the equation of a line passing through point $\left( 1,-1 \right)$ and parallel to PS which means that slope of the median PS is equal to the slope of the line. For that we require the slope of the median PS which we are going to calculate as if we have two points then we can write the slope of the line then the median PS will pass through point P and the centroid S. We know the coordinates of point P and centroid we can find by using the following formula which is equal to $\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)$ where ${{x}_{1}},{{x}_{2}},{{x}_{3}}$ are the x coordinates of the three vertices of triangle and ${{y}_{1}},{{y}_{2}},{{y}_{3}}$ are the y coordinates of the three vertices of triangle. The formula to find the slope from two points say $A\left( {{x}_{1}},{{y}_{1}} \right)\And B\left( {{x}_{2}},{{y}_{2}} \right)$ is equal to $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. Now, we know the slope of the line and a point passes through the line. We can write the equation of line as follows: $y+1=m\left( x-1 \right)$.

Complete step by step answer:

In the below diagram, we have shown a triangle PQR with PS as the median of this triangle:

We have to find the equation of line which is parallel to PS and passing through the point $\left( 1,-1 \right)$ so the slope of the required line is equal to the slope of the median.

To write the slope of the line we require two points so to write the slope of the median we require two points. One of the two points is P itself which we have given already and as the median is passing through the centroid of the triangle. We can calculate the centroid by using the following formula:
$\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)$
In the above formula, ${{x}_{1}},{{x}_{2}},{{x}_{3}}$ are the x coordinates of the three vertices of triangle and ${{y}_{1}},{{y}_{2}},{{y}_{3}}$ are the y coordinates of the three vertices of triangle
The x coordinates of P, Q and R are:
$2,6,7$
Now, adding all of them and dividing by 3 will give the x coordinate of centroid.
$\begin{align}
  & \dfrac{2+6+7}{3} \\
 & =\dfrac{15}{3}=5 \\
\end{align}$
The y coordinates of P, Q and R are:
$2,-1,3$
Now, adding all of them and dividing by 3 will give the y coordinate of centroid.
$\begin{align}
  & \dfrac{2-1+3}{3} \\
 & =\dfrac{4}{3} \\
\end{align}$
Hence, we got the coordinates of centroid as $\left( 5,\dfrac{4}{3} \right)$.
We have modified the above figure, by marking the centroid as G:

We know the formula of the slope when two points are given say $A\left( {{x}_{1}},{{y}_{1}} \right)\And B\left( {{x}_{2}},{{y}_{2}} \right)$ as:
$\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Now, we can find the slope of the median PS and we know the two points in which the median PS is passing through one is point P(2, 2) and the other is centroid that is $\left( 5,\dfrac{4}{3} \right)$. Substituting these points in the formula of slope we get,
$\begin{align}
  & \dfrac{\dfrac{4}{3}-2}{5-2} \\
 & =\dfrac{\dfrac{4-6}{3}}{3} \\
 & =-\dfrac{2}{9} \\
\end{align}$
Now, we know that the equation of a line with slope m and the point $\left( {{x}_{1}},{{y}_{1}} \right)$ it passes through is:
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Substituting the point as $\left( 1,-1 \right)$ and the slope as $-\dfrac{2}{9}$ in the above equation we get,
$y+1=\left( -\dfrac{2}{9} \right)\left( x-1 \right)$
Cross multiplying the above equation we get,
$\begin{align}
  & 9\left( y+1 \right)=\left( -2 \right)\left( x-1 \right) \\
 & \Rightarrow 9y+9=-2x+2 \\
 & \Rightarrow 2x+9y+7=0 \\
\end{align}$
Hence, we got the equation of line passing through $\left( 1,-1 \right)$ and parallel to PS as $2x+9y+7=0$.

So, the correct answer is “Option B”.

Note: Instead of writing the whole solution, the moment you get the slope of the desired line you can stop there and then compare the slope of all the equations given in the options. The slope of the option which matches with the slope of the desired line that we solved above is the right answer.
We are going to demonstrate what we just said.
The slope of the desired line that we got above is:
$-\dfrac{2}{9}$
Now, if we check slope of the option (a) we get,
We know the slope of the line $ax+by+c=0$ is equal to the negative of the division of coefficient of x and coefficient of y.
$-\dfrac{\text{coefficient of x}}{\text{coefficient of y}}$
(a)$4x-7y-11=0$
Slope of the above line is equal to:
$-\dfrac{4}{-7}=\dfrac{4}{7}$
As you can see that the above slope is not equal to the slope that we have calculated.
Similarly, you can check the remaining options.