Question

If the point $A(0,2)$ is equidistant from the points $B(3,p)$ and $C(p,5)$, find $\text{p}$. Also, find the length of $AB$.

Answer 1

Given,

$AB=AC$

$(AB{)}^2=(AC{)}^2$

Distance between two points=$\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

$(AB{)}^2=(AC{)}^2⟹$$(0-3{)}^2+(2-p{)}^2=(0-p{)}^2+(2-5{)}^2$

$9+4+p^2-4p=p^2+9$

$p=1$

$Distance=\sqrt{(0-3{)}^2+(2-1{)}^2}$

$Distance=\sqrt{10}$