Mathematico | Mathematico

The parabola

y = x^{2}

intersects the line $y = m x$ at $A$ and $y = - m x$ at $B$ .

Given that the area of the triangle $A B C$ is $8$ , find the value of $m .$

Hint

Express the coordinates of $A$ and $B$ in terms of $m .$ The area of the triangle is

\frac{1}{2} \times base \times height

Solution

Find an expression for $A$

\begin{aligned} m x & = x^{2} \\ x^{2} - m x & = 0 \\ x (x - m) & = 0 \\ x & \in {0, m} \\ A & = (m, m^{2}) \end{aligned}

By symmetry, $B = (- m, m^{2})$ .

The length $A B$ is $m - (- m) = 2 m$ , and the height of the triangle is $m^{2}$ (i.e. the $y$ coordinate of $A$ and $B$ ), so the area is given by

\begin{aligned} \frac{(2 m) (m^{2})}{2} & = 8 \\ m^{3} & = 8 \\ m & = 2 \end{aligned}