Coordinates and locus of centroid

Question

A triangle has two of its sides along the co-ordinate axes and its third side is a tangent to the circle $x^2+y^2=a^2$. If the coordinates of the point of contact of the tangent are $(a \cosØ,a \sinØ)$, show that the coordinates of the centroid are $(a/3 \cosØ, a/3 \sinØ)$. Show that the locus of the centroid is $1/x^2+1/y^2=9/a^2$

Answer 1

From the Article $#148$ of this, the tangent at $(a\cos\phi,a\sin\phi)$ is $x\cos\phi+y\sin\phi=a$

So, $$\frac x{\frac a{\cos\phi}}+\frac y{\frac a{\sin\phi}}=1$$

So,the vertices are $(0,0);(\frac a{\cos\phi},0);(0,\frac a{\sin\phi})$

So, the centorid will be $$\left(\frac{0+0+\frac a{\cos\phi}}3,\frac{0+\frac a{\sin\phi}+0}3\right)$$

Now, eliminate $\phi$