Question

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with x-axis.

Answer 1

Using similar triangles principle, we can write
$\frac{Q}{9}=\frac{y_1}{3}$
$Q=3y_1$



Similarly, $p=\frac{x}{3}$
Point P(x,y)
So $OB=x+\frac{x}{3}$
OA= y+3y= 4y
using pythagorous theorem, we get
$(4y{)}^2+{\left(\frac{4x}{3}\right)}^2+{12}^2$
$\Rightarrow \frac{9y^2+x^2}{81}=1\Rightarrow x^2+9y^2=81$
$\frac{y^2}{9}+\frac{x^2}{81}=1$ is the equation of the ellipse.