Multiple choice

Point $O$ is the centre of the ellipse with major axis $AB$ and minor axis $CD$. Point $F$ is one focus of the ellipse.If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is $2$, then the product $(AB)(CD)$ is equal to

  1. $65$
  2. $52$
  3. $78$

  4. None of the above
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A Correct answer
Explanation

For an ellipse, OF = ae = 6. The triangle OCF has sides OC=b, OF=ae, and CF=a. The diameter of the inscribed circle is 2, so the radius r = 1. The area of triangle OCF is rs = 1 * (a+b+ae)/2. Also, area = 0.5 * b * ae. Equating these and using a^2 = b^2 + (ae)^2 leads to the result.

AI explanation

In the ellipse, point O is the center at the origin (0,0), F is a focus at (6, 0), and C is an endpoint of the minor axis at (0, b). The semi-major axis a and semi-minor axis b are related to the focal distance OF = c by the equation c^2 = a^2 - b^2. The incircle of triangle OCF has a diameter of 2, meaning its radius is 1. Using the formula for the inradius of a right triangle, r = (b + c - a) / 2, we substitute the known values to get 1 = (b + 6 - a) / 2, which simplifies to a - b = 4. Combining this with the Pythagorean relation b^2 + 6^2 = a^2 gives a^2 - b^2 = 36. Factoring the difference of squares yields (a - b)(a + b) = 36, so 4(a + b) = 36, which means a + b = 9. Solving the system of equations gives a = 6.5 and b = 2.5. The major axis AB = 2a = 13 and the minor axis CD = 2b = 5. The product (AB)(CD) is therefore 13 * 5 = 65.

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