Question

Assertion (A) : Radius is the smallest distance of a tangent from the centre of the circle.
Reason (R) : Radius is perpendicular to the tangent.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

To verify if the Reason is the correct explanation, read the two statements by connecting them with "because":
"Radius is the smallest distance of a tangent from the centre of the circle because the radius is perpendicular to the tangent."
Since the shortest distance from a point to a line is always the perpendicular distance, this makes logical and mathematical sense!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is an "Assertion-Reason" type question based on the fundamental geometric properties of "Circles" and tangent lines.
We need to individually evaluate the truth of both the Assertion (A) and the Reason (R), and then check if the Reason is the correct logical explanation for the statement made in the Assertion.

Step 2: Key Formula or Approach:
Let's analyze both statements geometrically:

• Assertion (A): "Radius is the smallest distance of a tangent from the centre of the circle."
Let $O$ be the center of the circle, and let line $L$ be a tangent to the circle at point $P$.
For any point $Q$ on the tangent line other than $P$, the point $Q$ lies strictly in the exterior of the circle.
Therefore, the distance $OQ \gt \text{Radius } OP$.
This means the radius $OP$ is indeed the shortest/smallest distance from the center $O$ to any point on the tangent line. Thus, Assertion (A) is True.

• Reason (R): "Radius is perpendicular to the tangent."
By the standard theorem of circles, the radius to the point of contact is perpendicular to the tangent line ($OP \perp L$). Thus, Reason (R) is True.

Step 3: Detailed Explanation:

• We know from coordinate geometry and vector mathematics that the shortest distance from a given point (the center $O$) to a straight line (the tangent line $L$) is always the perpendicular distance.

• Since the radius is perpendicular to the tangent line at the point of contact (as stated in Reason R), it represents this perpendicular distance.

• Consequently, this perpendicular distance (which is equal to the radius) must be the smallest distance from the center to any point on the tangent line (as stated in Assertion A).

• Thus, the perpendicularity of the radius to the tangent line is the exact mathematical reason why the radius is the smallest distance.

• This means both statements are true, and the Reason is the correct explanation of the Assertion.

Step 4: Final Answer:
Both statements are true and Reason (R) correctly explains Assertion (A), which corresponds to Option (A).