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: Tangent to the parabola:
For \(y^2 = 4x\) (with \(a = 1\)), the tangent with slope m is \(y = mx + \frac{1}{m}\), that is, \(m^2x - my + 1 = 0\).

Step 2: Condition for the circle:
The circle \((x-3)^2 + y^2 = 9\) has centre \((3, 0)\) and radius 3. The distance from the centre to the tangent must be 3:
\[ \frac{\left|3m + \frac{1}{m}\right|}{\sqrt{1 + m^2}} = 3 \Rightarrow 9m^2 + 6 + \frac{1}{m^2} = 9 + 9m^2 \Rightarrow m^2 = \frac{1}{3} \]

Step 3: Choose the tangent above the axis:
\(m = \pm\frac{1}{\sqrt3}\). The point of contact on the parabola is \(\left(\frac{1}{m^2}, \frac{2}{m}\right)\), which lies above the X-axis only when \(m > 0\). So \(m = \frac{1}{\sqrt3}\).
\[ y = \frac{x}{\sqrt3} + \sqrt3 \Rightarrow \sqrt3\,y = x + 3 \]
Option (C).

Final Answer:
The common tangent is \(\sqrt3\,y = x + 3\), option (C). \[ \boxed{\sqrt{3}\,y = x + 3} \]