Straight lines minimum area of triangle

Question

Consider points P(7, 1) and O = (0, 0). If S is a point on the line y = x and T is a point on the x-axis so that P is on the line segment ST, then how to find minimum possible area of ΔOST for x coordinate of S greater than 1.

Answer 1

The area is $A=\frac{1}{2}x h$, where $h$ is the abscissa of T.

Since S, P and T are aligned, we can write $h$ as a function of $x$;

$\frac{y-1}{0-1}=\frac{x-7}{h-7}$.

Since $y=x$, we have:

$h=\frac{6x}{x-1}$.

Substituting the value of $h$, in the equation of the area, we get:

$A=\frac{3x^{2}}{x-1}$.

This function has a minimum in $(2,12)$.