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Q.
The locus of a point equidistant from two points whose position vectors are $\vec{a}$ and $\vec{b}$, is
Vector Algebra
Solution:
Locus of the point $\vec{r}$ equidistant from two points is given by
$\left|\vec{r}-\vec{a}\right|^{2}=\left|\vec{r}-\vec{b}\right|^{2}$
$\Rightarrow \left|\vec{r}\right|^{2}+\left|\vec{a}\right|^{2}-2\left(\vec{r}\cdot\vec{a}\right)=\left|\vec{r}\right|^{2}+\left|\vec{b}\right|^{2}-2\left(\vec{r}\cdot\vec{b}\right)$
$\Rightarrow 2\vec{r}\cdot\left(\vec{b}-\vec{a}\right)=\left|\vec{b}\right|^{2}-\left|\vec{a}\right|^{2}$
$\Rightarrow \vec{r}\cdot\left(\vec{b}-\vec{a}\right)=\frac{1}{2}\left\{\left(\vec{b}+\vec{a}\right)\cdot\left(\vec{b}-\vec{a}\right)\right\}$
$\Rightarrow \left\{\vec{r}-\frac{1}{2}\left(\vec{b}+\vec{a}\right)\right\}. \left(\vec{b}-\vec{a}\right)=0$