Question

The radius of the circle passes through the foci of a conic \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \) and has its centre at \( (0, 3) \), then the diameter of the circle is ---

• A. 7 units
• B. \( 2 \sqrt{12} \) units
• C. 8 units
• D. 4 units

Hint:

For an ellipse, the foci are located at a distance of \( c = \sqrt{a^2 - b^2} \) from the center along the major axis. The circle passing through the foci has its radius equal to the distance from the center to the foci.

Answer 1

The Correct Option is C

Step 1: Equation of the ellipse.
The given equation of the conic is:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
This is the equation of an ellipse with the center at the origin \( (0, 0) \), semi-major axis \( a = 4 \), and semi-minor axis \( b = 3 \).

Step 2: Foci of the ellipse.
For an ellipse, the foci are located at a distance of \( c \) from the center along the major axis, where:
\[ c = \sqrt{a^2 - b^2} \]
Substitute the values of \( a \) and \( b \):
\[ c = \sqrt{16 - 9} = \sqrt{7} \]
Thus, the foci are at \( (\pm \sqrt{7}, 0) \).

Step 3: Location of the circle.
The center of the circle is at \( (0, 3) \), and it passes through the foci of the ellipse. Since the foci are at \( (\pm \sqrt{7}, 0) \), the distance from the center of the circle at \( (0, 3) \) to each focus is the radius of the circle.
The distance between the center \( (0, 3) \) and a focus \( (\sqrt{7}, 0) \) is given by the distance formula:
\[ \text{Distance} = \sqrt{(\sqrt{7} - 0)^2 + (0 - 3)^2} = \sqrt{7 + 9} = \sqrt{16} = 4 \]

Step 4: Diameter of the circle.
The radius of the circle is 4 units, so the diameter is: \[ \text{Diameter} = 2 \times 4 = 8 \, \text{units} \]

Step 5: Conclusion.
Therefore, the diameter of the circle is 8 units, and the correct answer is option (C).