The Sliding Right Angle

#SlidingRightAngle#CoordinateGeometry#SLLoney

Problem Statement

A right-angled triangle $ABC$, having $C$ a right angle, is of given magnitude, that is $BC=a$ and $CA=b$. The angular points $A$ and $B$ slide along the positive $x$ and positive $y$ axes respectively. Find the locus of the vertex $C$.

Say that the coordinates of $C$ are $(h,k)$. Let $O$ be the origin $(0,0)$. Since $\angle AOB + \angle ACB = \pi $, therefore the quadrilateral $OACB$ is cyclic. Therefore, we have $\angle AOC = \angle ABC $.

$$ \implies \tan (\angle AOC) = \tan (\angle ABC) \tag{A} $$

Notice that $ \tan(\angle AOC) $ is simply the slope of the line $OC$, which is equal to $\frac{k}{h}$, and in the right triangle $ABC$, we have $ \tan (\angle ABC) = \frac{AC}{BC} = \frac{b}{a}. $ Therefore, using $(A)$ we have

$$ \frac{k}{h} = \frac{b}{a} \implies y =\frac{bx}{a}. $$

Which is the required locus.