Question

The radius of the circle represented by the equation 16(x² + y²) + 24x + 32y - 9 = 0 is?
1. 8
2. 9
3. \(\sqrt{\frac{17}{8}}\)
4. None of these.

Answer 1

Correct Answer - Option 3 : \(\sqrt{\frac{17}{8}}\)

Concept:

The radius of the circle represented by the general equation x2 + y2 + 2gx + 2fy + c = 0 is given by r² = g² + f² - c > 0, with the center at (-g, -f).

Calculation:

The given equation of the circle can be written as:

16(x2 + y2) + 24x + 32y - 9 = 0

⇒ x2 + y2 + \(\frac32\)x + 2y - \(\frac9{16}\) = 0

Comparing this with the general form of the circle's equation, we have:

g = \(\frac34\), f = 1, c = \(-\frac3{16}\)

Now, radius = \(\rm\sqrt{g^2+f^2-c}\)

= \(\rm\sqrt{\left(\frac34\right)^2+1^2-\left(-\frac{9}{16}\right)}\)

= \(\rm\sqrt{\frac9{16}+1+\frac{9}{16}}\)

= \(\sqrt{\frac{17}{8}}\)