The centre of a circle $C$ is at the centre of the ellipse $E:\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $\triangle PF_1F_2$ is $30$ and the length of the major axis of $E$ is $17$, then the distance between the foci of $E$ is:

Step 1: Identify key properties of the ellipse and circle

For an ellipse given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a>b$, the foci are located at $F_1 = (-ae, 0)$ and $F_2 = (ae, 0)$, where $e$ is the eccentricity. The center of the ellipse is at the origin $(0{,}0)$. The problem states that the circle $C$ is centered at the origin and passes through the foci $F_1$ and $F_2$. Therefore, the radius of the circle $C$ is the distance from the origin to either focus, which is $ae$.

Step 2: Use the definition of an ellipse

A fundamental property of an ellipse is that for any point $P$ on the ellipse, the sum of the distances from $P$ to the two foci $F_1$ and $F_2$ is constant and equal to the length of the major axis, $2a$. We are given that the length of the major axis is $17$, so $2a = 17$.

Step 3: Calculate the area of triangle $PF_1F_2$

The base of the triangle $\triangle PF_1F_2$ is the distance between the foci, which is $F_1F_2 = 2ae$. The height of the triangle is the absolute value of the $y$ -coordinate of point $P$, denoted as $|y_P|$, since $F_1$ and $F_2$ lie on the $x$ -axis. We are given that the area of $\triangle PF_1F_2$ is $30$.

Step 4: Relate point P to the circle and ellipse equations

Since point $P(x_P, y_P)$ lies on the circle $C$ with center $(0{,}0)$ and radius $ae$, its coordinates satisfy $x_P^2 + y_P^2 = (ae)^2$. Also, since $P$ lies on the ellipse, its coordinates satisfy the ellipse equation $\frac{x_P^2}{a^2} + \frac{y_P^2}{b^2} = 1$. We know that $b^2 = a^2(1-e^2)$.

Step 5: Solve for $y_P^2$

Substitute $x_P^2$ from the circle equation into the ellipse equation. Simplify the expression using $b^2 = a^2(1-e^2)$ and $a^2-b^2 = a^2e^2$. This allows us to solve for $y_P^2$ in terms of $a$, $b$, and $e$.

Step 6: Substitute $y_P$ into the area equation and solve for $ae$

Now substitute the expression for $|y_P|$ into the area equation $ae|y_P| = 30$. This simplifies directly to $b^2 = 30$.

Step 7: Calculate the distance between the foci

We have $2a = 17$, so $a = \frac{17}{2}$. We found $b^2 = 30$. The distance between the foci is $2ae$. We use the relationship $a^2e^2 = a^2 - b^2$ to find $ae$. Substitute the values of $a$ and $b^2$ to calculate $ae$, and then multiply by 2 to get the final distance.

More from this chapter

• Find the equation of the circle passing through points (1,2), (3,4), and (5,2).
• Determine the equation of the circle with radius 4 and Centre (-2, 3).
• Find the equation of the hyperbola where foci are (0, +-6) and the length of the latus rectum is 36.
• Find the equation of the hyperbola where foci are (0, ± 12) and the length of the latus rectum is 36
• Find the equation of the circle with centre (-3,2) and radius 10.
• Find the equation of the circle with centre (-3,2) and radius 9.
• Find the equation of the circle with centre (-3,2) and radius 8.
• Find the equation of the circle with centre (-3,2) and radius 6.

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