Question

The radius of the circle passing through the foci of the ellipse \(9x^2 + 16y^2 = 144\) and having its centre at \((0,3)\), is

• A. 4
• B. 3
• C. \(\sqrt{12}\)
• D. \(\frac{7}{2}\)

Hint:

For ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a>b\), foci are \((\pm ae, 0)\) where \(e = \sqrt{1 - b^2/a^2}\).

Answer 1

The Correct Option is A

To solve this problem, we need to find the radius of a circle that passes through the foci of the ellipse given by the equation \(9x^2 + 16y^2 = 144\) and has its center at \((0, 3)\). 

1. First, let's identify the standard form of the ellipse equation. The given equation is: \(9x^2 + 16y^2 = 144\) Divide the entire equation by 144: \(\frac{9x^2}{144} + \frac{16y^2}{144} = 1\) Simplifying: \(\frac{x^2}{16} + \frac{y^2}{9} = 1\)
2. This is the standard form of an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) With \(a^2 = 16 \Rightarrow a = 4\) and \(b^2 = 9 \Rightarrow b = 3\)
3. For this ellipse, \(a > b\), so it is horizontally oriented, and the foci are located at \((\pm c, 0)\), where \(c = \sqrt{a^2 - b^2} = \sqrt{16 - 9} = \sqrt{7}\). Thus, the foci are at points \((\pm \sqrt{7}, 0)\).
4. We are given that the center of the circle is \((0, 3)\). The distance from the center of the circle to either focus is the radius of the circle. Calculate the distance using the distance formula: \(r = \sqrt{(0 - \sqrt{7})^2 + (3 - 0)^2} = \sqrt{7 + 9} = \sqrt{16} = 4\)
5. Thus, the radius of the circle is 4. Therefore, the correct option is:

4 
 

This step-by-step method calculates the radius of the circle as required by identifying correct elements of the ellipse, solving for foci, and using the distance formula. The reasoning is verified logically and mathematically, leading to option 4 as the correct answer.