Question

The equation of the tangent to the curve given by $x=a \sin^{3}t$, $y=b \cos^{3}t$ at a point where $t = \frac{\pi}{2}$ is ________.

• A. $y=1$
• B. $y=0$
• C. $x=0$
• D. $x=1$

Hint:

If the slope is zero, the tangent is a horizontal line ($y = k$).

Answer 1

The Correct Option is B

Step 1: Concept
Find coordinates $(x, y)$ and slope $dy/dx$ at $t = \pi/2$.
Step 2: Analysis
At $t = \pi/2$:
$x = a \sin^3(\pi/2) = a(1) = a$.
$y = b \cos^3(\pi/2) = b(0) = 0$.
Step 3: Slope Calculation
$dx/dt = 3a \sin^2 t \cos t$.
$dy/dt = -3b \cos^2 t \sin t$.
$dy/dx = \frac{-b \cos t}{a \sin t}$. At $t = \pi/2$, slope $= 0$.
Step 4: Conclusion
Equation: $y - 0 = 0(x - a) \Rightarrow y = 0$.
Final Answer:(B)