Question

In \(△ABC\), with usual notation, if cot A, cot B, cot C are in arithmetic progression, then

• A. sin A, sin B, sin C are in arithmetic progression.
• B. \(a^2,b^2,c^2\) are in arithmetic progression.
• C. cos A, cos B, cos C are in arithmetic progression.
• D. a, b, c are in arithmetic progression.

Hint:

A tangent to the parabola is y = mx + 1/m; set its distance from the circle centre equal to the radius.

Answer 1

The Correct Option is B

Step 1: Check the options directly:
Options (A) and (D) give $y = \sqrt3 x + \ldots$ lines with slope $\pm\sqrt3$. Convert (C): $x - \sqrt3 y + 3 = 0$, slope $\frac{1}{\sqrt3}$.

Step 2: Verify both tangencies:
Parabola: with $m = \frac{1}{\sqrt3}$, the intercept should be $\frac{1}{m} = \sqrt3$. The line has intercept $\frac{3}{\sqrt3} = \sqrt3$. Matches.
Circle: distance of $(3, 0)$ from $x - \sqrt3 y + 3 = 0$ is $\frac{|3 + 3|}{\sqrt{1 + 3}} = \frac{6}{2} = 3$, equal to the radius. So (C) is a common tangent.

Final Answer:
$\sqrt3\,y = x + 3$. \[ \boxed{\sqrt{3}\,y = x + 3} \]