Answer to Question #250669 in Analytic Geometry for Aliyah

Question

Find the equation of the locus of the third vertex of a triangle if two of its vertices are (1, -2) and (5, 0) and whose area is 3.

Answer 1

Use Determinant method ,

A= $\pm\frac{1}{2}\begin{bmatrix} x & y & 1\\ 1& -2& 1\\ 5& 0& 1 \end{bmatrix}$ =3

x(-2-0)-y(1-5)+(0-(-10)= $\pm6$

-2x+4y+10=$\pm6$

Hence 4y= 2x-10+6 or 4y= 2x-10-6

that is , 4y = 2x -4 or 4y = 2x -16

y=1/2x-1 or y= 1/2x-4.

and the required locus is any point on either of these lines.