Question

Let a=(2,5) and b=(4,-1) are the two vertices of a triangle abc .third vertex c moves along l=9x+7y+4=0.the locus of the centroid of trisngle abc is the line

Answer 1

this will help you then real is 9x+7y=26

Answer 2

The locus of the centroid of the triangle ABC is 9x + 7y = 26.

Step-by-step explanation:

Given that, the coordinates of A and B are (2, 5) and (4, - 1) respectively.

Let the vertex C has coordinates (p, q). So the point C satisfies the equation 9x + 7y + 4 = 0.

Then,

9p + 7q + 4 = 0 ..... (i)

Let, (x₁, y₁) be the centroid of the triangle ABC.

Then, x₁ = (2 + 4 + p)/3, y₁ = (5 - 1 + q)/3

i.e., x₁ = (6 + p)/3, y₁ = (4 + q)/3

or, p = 3x₁ - 6, q = 3y₁ - 4

From (i), substituting p = 3x₁ - 6 and q = 3y₁ - 4, we get

9 (3x₁ - 6) + 7 (3y₁ - 4) + 4 = 0

or, 27x₁ - 54 + 21y₁ - 28 + 4 = 0

or, 27x₁ + 21y₁ = 78

or, 9x₁ + 7y₁ = 26

∴ the locus of the centroid is given by

9x + 7y = 26