Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
The equation of a circle of radius $1$ touching the circles $x^2 + y^2 - 2|x| = 0$ is:
$x^2 + y^2 + 2\sqrt{3}y - 2 = 0$
$x^2 + y^2 - 2\sqrt{3}y + 2 = 0$
$x^2 + y^2 + 2\sqrt{3}y + 2 = 0$
$x^2 + y^2 - 2\sqrt{3}y - 2 = 0$
Step-by-Step Solution
Key Concept: Two circles tangent to two given intersecting circles have centers equidistant from the line joining the given centers, found using the tangency condition.
The two given circles are $x^2 + y^2 - 2x = 0$ (center at $(1,0)$) and $x^2 + y^2 + 2x = 0$ (center at $(-1,0)$). From the figure, the centres of the required circles are at $(0, \sqrt{3})$ and $(0, -\sqrt{3})$. Both circles have radius $1$, so their equations are $(x-0)^2 + (y \mp \sqrt{3})^2 = 1$.
Correct Answer: 2,3