Coordinate Geometry
Circle inscribed in ellipse; radius range
MJMT_Full_Test_02
Grade 12
Question:
If a circle is inscribed in ellipse $x^2+4y^2=4$, then the range of radius of the circle is
$[0,1)$
$(0,1]$
$\left[\frac{1}{2},1\right]$
$\left(\frac{1}{2},1\right]$
Step-by-Step Solution
Key Concept: The inscribed circle has its center on the major axis. If the circle touches the ellipse at a point, the normal at that point passes through the center.
Radius $r^2=1-\frac{3}{4}\cos^2\theta \in [\frac{1}{4},1]$, so $r\in[\frac{1}{2},1]$.
Correct Answer: 3