Question

In a triangle $\Delta ABC$, A (2, 3) and medians through B and C have equations x + y – 1 = 0 and 2y – 1 = 0. Equation of median through A is:
a). x + y = 4
b). 5x - 3y = 1
c). 5x + 3y = 1
d). 5x = 3y

Answer 1

Hint: Firstly, find the centroid of $\Delta ABC$. The centroid of $\Delta ABC$ is the intersection of the medians. So, find the intersection of the two medians and the point A is already given in the question. And from the two points, we can find the equation of a line.
From the below figure you can better visualize the question:

Complete step-by-step solution -

BD and CE are the medians and F is the centroid of $\Delta ABC$. Now, we are going to find the intersection of these medians.
x + y – 1 = 0………..Eq. (1)
2y – 1 = 0……………Eq. (2)
Solving the above two equations by substitution method. From Eq. (2) we will get the value of $y=\dfrac{1}{2}$ the substituting this value of y in Eq. (1) we get,
$\begin{align}
  & x+\dfrac{1}{2}-1=0 \\
 & \Rightarrow x-\dfrac{1}{2}=0 \\
 & \Rightarrow x=\dfrac{1}{2} \\
\end{align}$
So, we have got the intersection of these lines as $\left( \dfrac{1}{2},\dfrac{1}{2} \right)$ which is the centroid.
Now, we have a point A (2, 3) and centroid $F\left( \dfrac{1}{2},\dfrac{1}{2} \right)$ so we can write the equation of a line passing through these two points.
For writing the equation of a line, we need a slope and a point. So, we are going to find the slope of a line.
The formula for slope of the line passing through two points $ P (x_1, y_1) $ and $ Q(x_2, y_2) $ is as follows:

Here, we are considering point A as Q and point F as P. Now, substituting the values of point A (2, 3) and point $F\left( \dfrac{1}{2},\dfrac{1}{2} \right)$ in the above equation in “m” we get,
$\begin{align}
  & m=\dfrac{3-\dfrac{1}{2}}{2-\dfrac{1}{2}} \\
 & \Rightarrow m=\dfrac{5}{3} \\
\end{align}$
As we have got the slope of the equation and we can take any point from A and F. So, we are taking point A (2, 3). Now, we can write an equation passing through A as we have a point A and the slope.
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Substituting the value of m and $ (x_1, y_1) $ as (2, 3) in the above equation we get:
$\begin{align}
  & y-3=\dfrac{5}{3}\left( x-2 \right) \\
 & \Rightarrow 3y-9=5x-10 \\
 & \Rightarrow 5x-3y=1 \\
\end{align}$
Hence, the equation of the median passing through A (2, 3) is 5x – 3y = 1.
Hence, the correct option is (b).

Note: While choosing a point from one of the two points for writing the equation of a straight line, always choose a point which is not in fraction or having 0 in one of the coordinates as it will reduce the calculation complexity.