Analytic of point

Question

I have a Problem with analytic of point A(1/2,-3) and B(a,2a) are given if $AB^2$=3/4 find the sum of possible values of a
How can we square B ? Is $AB^2$ a line or something?

Answer 1

you have $$\vec{AB}=\left[a-\frac{1}{2};2a+3\right]$$ so we get $$(\vec{AB})^2=\left(a-\frac{1}{2}\right)^2+(2a+3)^2=\frac{3}{4}$$ this is the dot product and must be solved for $a$ $$\vec{AB}=[x_B-x_A;y_B-y_A]$$