The point $A$ is the vertex of the parabola

The point $B$ lies on the parabola to the right of $A.$ The point $M$ is the midpoint of $AB.$

A triangle $PQR$ is formed, where

• $P,Q$ are the points of intersection between the parabola and the horizontal line through $M$

• $R$ lies on the parabola directly below $M.$

Given that the area of the triangle $PQR$ is $\sqrt{2},$ find the coordinates of $B.$

Your diagram should look like this:

$A$ has coordinates $(1,1).$ If we let $B$ have $x$-coordinate $b,$ then we have $B=(b,b^2-2b+2).$

The coordinates of $M$ are

$R$ has the same $x$-coordinate as $M$

$P,Q$ have the same $y$-coordinate as $M.$

After finding $R$ in terms of $b$, you should find that the height of the triangle is

After finding $P$ and $Q$ in terms of $b$, you should find that the base of the triangle is