Question

$B$ and $C$ are fixed points having co-ordinates $(3,0)$ and $(-3,0)$, respectively. If the vertical $\angle BAC$ is ${90}^o$, then the locus of the centroid of the $\Delta ABC$ has the equation

• A. $x^2+y^2=1$
• B. $x^2+y^2=2$
• C. $9(x^2+y^2)=1$
• D. $9(x^2+y^2)=4$

Answer 1

The correct option is A $x^2+y^2=1$

Let the third vertex A be $(a,b)$ and the centroid G be $(h,k)$
Since $\angle BAC={90}^o$, A lies on the circle which has BC as the diameter.
The equation of the circle is :
$x^2+y^2=9$
$\Rightarrow a^2+b^2=9$ - (i)
The centroid of $\Delta ABC=(\frac{-3+3+a}{3},\frac{0+0+b}{3})=(\frac{a}{3},\frac{b}{3})$
Substituting $x=\frac{a}{3}$ , $y=\frac{b}{3}$ in (i)
$x^2+y^2=1$