Question

The minimum area of the triangle formed by the variable line $3cos\theta \cdot x+4sin\theta \cdot y=12$ and the co-ordinate axes is _____.

• A. $144$
• B. $\frac{25}{2}$
• C. $\frac{49}{4}$
• D. $12$

Answer 1

Answer: (D)

$12$

The line $3cos\theta .x+4sin\theta .y-12=0$ intersects the $x$ axis at $(\frac{4}{\cos \theta },0)$ and $y$ axis at $(0,\frac{3}{sin\theta })$

Thus, the area of the formed right angled triangle would be $\frac{1}{2}\times$ height $\times$ base $=\frac{1}{2}\times \frac{4}{cos\theta }\times \frac{3}{sin\theta }$ $=\frac{12}{sin2\theta }$

Since we need the minimum area possible, we maximize the denominator and thus, since the maximum value of $sin2\theta$ can be 1, the minimum area would be $12$