Question

The curves C₁ : y = 1 - cosx, x ∈ (0, π) and C₂: y = $\frac{\sqrt{3}}{2}$|x| + a will touch each other if

• A. a = $\frac{3}{2} - \frac{\pi}{\sqrt{3}}$
• B. a = $\frac{3}{2} - \frac{\pi}{2\sqrt{3}}$
• C. a =$\frac{1}{2} - \frac{\pi}{\sqrt{3}}$
• D. a = $\frac{3}{4} - \frac{\pi}{\sqrt{3}}$

Answer 1

Answer: (A)

a = $\frac{3}{2} - \frac{\pi}{\sqrt{3}}$

Answer 2

Slope of C₁ is sinx and for x > 0 slope of C₂ is $\frac { \sqrt { 3 } } { 2 }$. Thus for x = $\frac { \sqrt { 3 } } { 2 }$ ⇒ x = $\frac { \pi } { 3 }$ or $\frac { 2 \pi } { 3 }$

Hence point of contact is $\left( \frac { \pi } { 3 } , \frac { 1 } { 2 } \right)$ or $\left( \frac { 2 \pi } { 3 } , \frac { 3 } { 2 } \right)$

For $\left( \frac { \pi } { 3 } , \frac { 1 } { 3 } \right)$ we get $a = \frac { 1 } { 2 } - \frac { \pi } { 2 \sqrt { 3 } }$

For $\left( \frac { 2 \pi } { 3 } , \frac { 3 } { 2 } \right)$ we get