Question

If the circle \[ x^2+y^2+2ax+c=0 \] lies completely inside the circle \[ x^2+y^2+2bx+c=0, \] then which of the following holds?

• A. \(ab\lt 0\)
• B. \(c\lt 0\)
• C. \(c=0\)
• D. \(ab\gt 0\)

Hint:

For one circle to lie completely inside another: \[ \text{distance between centres}+\text{smaller radius}\lt \text{larger radius} \] Always compare both centres and radii carefully.

Answer 1

The Correct Option is D

Step 1: Find the centres and radii of the circles.
For the circle \[ x^2+y^2+2ax+c=0, \] the centre is \[ (-a,0) \] and radius is \[ r_1=\sqrt{a^2-c} \] For the circle \[ x^2+y^2+2bx+c=0, \] the centre is \[ (-b,0) \] and radius is \[ r_2=\sqrt{b^2-c} \]

Step 2: Use the condition for one circle lying completely inside another.
If the first circle lies completely inside the second circle, then \[ \text{distance between centres}+r_1\lt r_2 \] So, \[ |a-b|+\sqrt{a^2-c}\lt \sqrt{b^2-c} \]

Step 3: Compare the radii.
Since \[ \sqrt{a^2-c}\lt \sqrt{b^2-c}, \] we get \[ a^2-c\lt b^2-c \] \[ a^2\lt b^2 \] Thus, \[ |a|\lt |b| \]

Step 4: Use the centre-distance condition.
The distance between centres is \[ |a-b| \] For the smaller circle to remain completely inside the larger circle, both centres must lie on the same side of the origin.
Hence, \[ a \text{ and } b \] must have the same sign.

Step 5: Interpret same sign condition.
If two numbers have the same sign, then their product is positive.
Therefore, \[ ab\gt 0 \]

Step 6: Reject other options.
There is no necessity that \[ c\lt 0 \] or \[ c=0 \] Also, \[ ab\lt 0 \] would mean the centres lie on opposite sides, which contradicts the containment condition.

Step 7: Final conclusion.
Hence, \[ \boxed{ab\gt 0} \]