Question

If the intercepts made by a variable circle on the X-axis and Y-axis are 8 and 6 units respectively, then the locus of the center of the circle is:

• A. \( x^2 - y^2 + 28 = 0 \)
• B. \( y^2 - x^2 - 7 = 0 \)
• C. \( x^2 - y^2 - 28 = 0 \)
• D. \( x^2 - y^2 - 7 = 0 \)

Hint:

For locus problems, always express geometric conditions as equations and eliminate parameters systematically.

Answer 1

The Correct Option is D

Step 1: General Equation of the Circle
A circle with center \( (h, k) \) and radius \( r \) has the equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] Step 2: X-Intercept Condition
For X-intercepts (y=0): \[ 2\sqrt{r^2 - k^2} = 8 \implies r^2 - k^2 = 16 \quad (1) \] Step 3: Y-Intercept Condition
For Y-intercepts (x=0): \[ 2\sqrt{r^2 - h^2} = 6 \implies r^2 - h^2 = 9 \quad (2) \] Step 4: Eliminate \( r^2 \)
Subtract (2) from (1): \[ h^2 - k^2 = 7 \implies x^2 - y^2 = 7 \] Conclusion:
The correct option is (4).