Question

Let the points $P_1(\frac{\pi}{4}), P_2(\frac{3\pi}{4}), P_3(\frac{5\pi}{4}), P_4(\frac{7\pi}{4})$ lie on the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$. Then they form:

• A. a rectangle
• B. a square
• C. a parallelogram
• D. a rhombus

Hint:

Parametric Points on Hyperbola:

• Standard form: $(a\sec\theta, b\tan\theta)$
• Points $(\pm x, \pm y)$ lie at symmetric rectangle corners.
• Check distances and orientation for shape classification.

Answer 1

The Correct Option is A

Use parametric form of hyperbola: $x = 3\sec\theta, y = 4\tan\theta$ Evaluating:

• $\theta = \frac{\pi}{4} \Rightarrow x = 3\sqrt{2}, y = 4$
• $\theta = \frac{3\pi}{4} \Rightarrow x = -3\sqrt{2}, y = -4$
• $\theta = \frac{5\pi}{4} \Rightarrow x = -3\sqrt{2}, y = 4$
• $\theta = \frac{7\pi}{4} \Rightarrow x = 3\sqrt{2}, y = -4$

Points formed: $(\pm 3\sqrt{2}, \pm 4)$ $\Rightarrow$ axes-aligned rectangle.