Question

Let A(2, - 3) and B(-2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line \[2x + 3y = 1\] , then the locus of the vertex C is the line
A. 3x – 2y = 3
B. 2x + 3y = 9
C. 2x – 3y = 7
D. 3x + 2y =5

Answer 1

Hint: We are given a triangle ABC with points of A and B. Also, given a centroid that lies on a line of the equation \[2x + 3y = 1\] . We need to find the line on which the point C lies. For this, we will draw the diagram from the given information. And then, we will use the formula of the equation of the centroid of the triangle to get the coordinates of C by substituting the values of vertex A and B. Thus, we will substitute this point value in the given line to get the final output.

Complete step by step answer:
Given that, a triangle ABC has 2 vertices A and B. Let \[A\left( {2, - 3} \right) = \left( {{x_1},{y_1}} \right)\] and \[B\left( { - 2,1} \right) = \left( {{x_2},{y_2}} \right)\]
And let point \[C = \left( {{x_3},{y_3}} \right)\]
As we know, the centroid is the centre point of the object.
We are also given that a centroid lies on the line \[2x + 3y = 1\].

According to the given information, we will draw a diagram as below:

Here, point O is the centroid of the triangle ABC.We know that, the equation of the centroid of the triangle is,
\[(\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_2} + {y_3}}}{3})\] ----- (1)

Thus, the coordinates of the triangle ABC is as below:
Substituting the values of A and B in the equation (1), we will get,
\[(\dfrac{{2 - 2 + {x_3}}}{3},\dfrac{{ - 3 + 1 + {y_3}}}{3})\]
\[\Rightarrow (\dfrac{{{x_3}}}{3},\dfrac{{ - 2 + {y_3}}}{3})\]
Since, we are given that the centroid lies on the line
\[2x + 3y = 1\]
Substituting the above values in this equation, we will get,
\[ \Rightarrow 2(\dfrac{{{x_3}}}{3}) + 3(\dfrac{{ - 2 + {y_3}}}{3}) = 1\]

Taking LCM as 3 and removing the brackets, we will get,
\[ \Rightarrow 2{x_3} - 6 + 3{y_3} = 3\]
\[ \Rightarrow 2{x_3} + 3{y_3} - 6 = 3\]
By using transposition, we will move the RHS term to LHS, we will get,
\[ \Rightarrow 2{x_3} + 3{y_3} - 6 - 3 = 0\]
\[ \Rightarrow 2{x_3} + 3{y_3} - 9 = 0\]
Again by using transposition, we will get,
\[ \Rightarrow 2{x_3} + 3{y_3} = 9\]
\[ \therefore 2x + 3y = 9\]

Hence, the locus of the vertex C is the line \[2x + 3y = 9\].

Note: The centroid of a triangle is formed when three medians of a triangle intersect. The point of intersection of the medians of a triangle is known as centroid. The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. The orthocentre is the intersection point of the altitudes whereas, the centroid is the intersection point of the medians.