In the diagram, two circles with centres $A$ and $B$ intersect at points $P$ and $Q$ so that $\angle P A Q=60^{\circ}$ and $\angle P B Q=90^{\circ}$. What is the ratio of the area of the circle with centre $A$ to the area of the circle with centre $B$ ?
(A) 3:1
(B) $3: 2$
(C) 4:3
(D) $2: 1$
(E) $9: 4$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-124.jpg?height=206&width=317&top_left_y=1778&top_left_x=1338)
Solution 1

Join $A$ to $B$ (meeting $P Q$ at $X$ ) and consider the triangle $A P B$.

By symmetry, $\angle P A B=30^{\circ}, \angle P B A=45^{\circ}$ and $P X$ is perpendicular to $A B$.

Since $A P=R$ and $\triangle A P X$ is a $30-60-90$ triangle, then

$P X=\frac{1}{2} R$.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-205.jpg?height=274&width=545&top_left_y=291&top_left_x=1365)

Since $\triangle B P X$ is a $45-45-90$ triangle, then $P B=\sqrt{2}\left(\frac{1}{2} R\right)$.

Therefore, $r=\sqrt{2}\left(\frac{1}{2} R\right)$ or $r^{2}=\frac{1}{2} R^{2}$.

Since the two circles have areas of $\pi r^{2}$ and $\pi R^{2}$, then the ratio of their areas is $2: 1$.

## Solution 2

Suppose that the radius of the circle with centre $A$ is $R$, and the radius of the circle with centre $B$ is $r$. Join $P$ to $Q$.

Since $P Q$ is a chord in each of the two circles, we will try to find the length of $P Q$ in terms of $R$ and in terms of $r$, in order to find a relationship between $R$ and $r$.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-205.jpg?height=271&width=436&top_left_y=916&top_left_x=1365)

Consider $\triangle A P Q$. Then $A P=A Q=R$ and $\angle P A Q=60^{\circ}$, so $\triangle A P Q$ is isosceles and has an angle of $60^{\circ}$, so must be equilateral. Therefore, $P Q=R$, or $P Q^{2}=R^{2}$.

Now looking at $\triangle B P Q$. Then $B P=B Q=r$, and $\angle P B Q=90^{\circ}$, so $P Q^{2}=r^{2}+r^{2}=2 r^{2}$, by Pythagoras.

So we know that $P Q^{2}=R^{2}=2 r^{2}$, or $\pi R^{2}=2\left(\pi r^{2}\right)$.

But the expression on the left side is the area of the circle with centre $A$ and the expression in the parentheses on the right side is the area of the circle with centre $B$. Thus, the required ratio is $2: 1$, since the area of the left-hand circle is twice the area of the right-hand circle.

ANSWER: (D)

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