Question

what is the vertex and radius of (x-3)^2 +(x-5)^2=100

Answer 1

Vertex is at $V(3,5)$ and radius is $10$.

Lets try to grasp on those numbers.

We know that the distance between two different points $A(x_1,y_1),B(x_2,y_2)$ on a two-dimensional plane is defined by this formula,

$d(A,B)=\sqrt[2]{(x_2-x_1)^2+(y_2-y_1)^2}$.

Now imagine a circle with its vertex at point $V(u,v)$ and a point lying on the circumference of this circle $P(x,y)$.

If we want to know the distance between these two points (radius) we use the general formula,

$r=\sqrt[2]{(x-u)^2+(y-v)^2}$

Squaring both sides gives us,

$r^2=(x-u)^2+(y-v)^2$.

So what we are given the corresponding form to your equation.

In your case $r^2=100\implies r=10$.

And $u=3, v=5$ which construct the vertex $V(u,v)=V(3,5)$.

Hope this helps.