Question

If $\tan\theta,\;2\tan\theta+2,\;3\tan\theta+3$ are in geometric progression and $\tan\theta\neq -1$, then $\tan\theta=$

• A. -3
• B. -4
• C. -6
• D. -8

Hint:

For three terms in G.P., always use the relation $(\text{middle term})^2=(\text{first term})(\text{third term})$.

Answer 1

The Correct Option is B

Concept: If three numbers are in G.P., then the square of the middle term equals the product of the first and third terms.

Step 1: Let $t=\tan\theta$.
Then the three terms are \[ t,\qquad 2t+2,\qquad 3t+3 \] Since they are in G.P., \[ (2t+2)^2=t(3t+3) \]

Step 2: Simplify the equation.
\[ 4(t+1)^2=3t(t+1) \] Given \(t\neq -1\), divide both sides by \((t+1)\): \[ 4(t+1)=3t \] \[ 4t+4=3t \] \[ t=-4 \] Hence, \[ \tan\theta=-4 \]