Question

Assertion :The circle of smallest radius passing through two given points $A$ & $B$ must be of radius $\frac{1}{2}AB$. Reason: A straight line is a shortest distance between two points.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Let $C_1$ be the circle passing through $A$ and $B$ whose diameter is $AB$

and $C_2$ be another circle which passes through $A$ and $B$ then center of $C_1$ and $C_2$ must lies on the $\perp$ bisector of $AB$

Indeed the center of $C_1$ is mid point $M$ of $AB$ and center of any other circle lies somewhere else on the bisector.
Then $M_{1}A>AM$ $($ hypotenuse of right angle. $\Delta AM{M}_1)$
$\Rightarrow$ Radius of $C_2>\frac{1}{2}AB$
$\Rightarrow C_1$ is the circle whose radius is the least.
Thus the Assertion is true but does not actually follow from reason which is certainly true.
Hence, choice (B) is correct answer.