2024/008) Find minimum Value of $(a+7)^2+(b+2)^2$ with Constraint $(a-5)^2+(b-7)^2=4$

We have the constraint  $(a-5)^2+(b-7)^2=4$ this is set of points lying in a circle with centre (5,7) and radius 2.

We need to find the minimum of  square of the distance of (a,b) from (-7,-2) and this is minimum when  (a,b) lies in the line from (5,7) to (-7,-2)

Distance from (5,7) to (-7,2) = $\sqrt{(5+7)^2 + (7+2)^2}= \sqrt{12^2 + 9^2} = 15$

So distance from (a,b) to (-7,  -2) is $15-2 = 13$

Minimum Value of $(a+7)^2+(b+2)^2= 13^2 =169$