Let the ellipse E:x^2+9y^2=9 intersect the positive x- and y-axes at points A and B respec…

Question

Let the ellipse $E:x^2+9y^2=9$ intersect the positive $x$- and $y$-axes at points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line through $A$ and $B$ meet circle $C$ at the point $P$. If the area of the triangle with vertices $A$, $P$, and the origin $O$ is $\dfrac mn$, with $m,n$ coprime, then $m-n$ is equal to:

Accepted Answer

$E:\dfrac{x^2}9+y^2=1$, so $A=(3,0),\ B=(0,1)$, and $C:x^2+y^2=9$.

Line $AB$: $x+3y=3$. Substituting $x=3-3y$ into $x^2+y^2=9$: $10y^2-18y=0\Rightarrow y=0\ (\text{point }A)$ or $y=\dfrac9{5}$, giving
$$P=\left(-\frac{12}5,\frac95\right).$$

$$\text{Area}(OAP)=\frac12\left|x_Ay_P-x_Py_A\right|=\frac12\left|3\cdot\frac95\right|=\frac{27}{10}.$$

$$m=27,\ n=10\ (\text{coprime}),\qquad m-n=17.$$

Solutions · 1