Mathematico | Mathematico

Two lines are said to be parallel if they have the same gradient.

The line segment connecting $A (a, α)$ with $B (4, 1)$ is parallel and equal in length to the line segment connecting $C (- 1, 3)$ with $D (4, 15) .$

Find the possible coordinates of the point $A .$

What kind of shape is $A B C D$ ?

By considering the gradient of $A B$ and $C D$ , you can show that

5 (α - 1) = 12 (a - 4)

By considering the lengths $| A B |$ and $| C D |,$ you can show that

(a - 4)^{2} + (α - 1)^{2} = 13^{2}

Now use the equation you got from the previous hint to replace $(α - 1)$ in this equation and solve to find $a .$