Question

A=(-2,1) B=(4,5) and c is a point on locus eqution in y square 4x find the equation of the locus of the centroid of ∆ABC
​

Answer 1

Answer:

The required locus is

$9y^2-12x-36y+44=0$

Step-by-step explanation:

Formula used:

Centroid of a triangle having vertices

$(x_1,y_1),\:(x_2,y_2)\:and\:(x_3,y_3)\:is$

$(\frac{x_1+x_2+x_3 }{3},\frac{y_1+y_2+y_3 }{3})$

Given points are A(-2,1) and B(4,5)

Let the co ordinates of the C be (a,b)

Let (h, k) be the centroid of ΔABC.

Then

$(\frac{x_1+x_2+x_3 }{3},\frac{y_1+y_2+y_3 }{3})=(h,k)$

$(\frac{-2+4+a}{3},\frac{1+5+b}{3})=(h,k)$

$(\frac{2+a}{3},\frac{6+b}{3})=(h,k)$

$\frac{2+a}{3}=h$

$2+a=3h$

$a=3h-2$

$\frac{6+b}{3}=k$

$6+b=3k$

$b=3k-6$

since (a,b) lies on $y^2=4x$

$(3k-6)^2=4(3h-2)$

$9k^2+36-36k=12h-8$

$9k^2-12h-36k+44=0$

Therefore, the locus of (h,k) is

$9y^2-12x-36y+44=0$