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.

Answer 2

Step 1: Understand the points and the hyperbola
Given points \( P_1\left(\frac{\pi}{4}\right), P_2\left(\frac{3\pi}{4}\right), P_3\left(\frac{5\pi}{4}\right), P_4\left(\frac{7\pi}{4}\right) \) lie on the hyperbola
\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]
These points are likely parametric points defined using a parameter \(t\).

Step 2: Parametric form of the hyperbola
For the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), a common parametric representation is:
\[ x = a \sec t, \quad y = b \tan t \] where \( a = 3 \) and \( b = 4 \).

Step 3: Calculate coordinates of each point
Calculate coordinates of \( P_i \) for each given \( t \):
- \( P_1\left(\frac{\pi}{4}\right): x = 3 \sec\frac{\pi}{4} = 3 \times \frac{\sqrt{2}}{1} = 3\sqrt{2} \),
\( y = 4 \tan\frac{\pi}{4} = 4 \times 1 = 4 \)
So, \( P_1 = (3\sqrt{2}, 4) \)

- \( P_2\left(\frac{3\pi}{4}\right): x = 3 \sec\frac{3\pi}{4} = 3 \times \left(-\sqrt{2}\right) = -3\sqrt{2} \),
\( y = 4 \tan\frac{3\pi}{4} = 4 \times (-1) = -4 \)
So, \( P_2 = (-3\sqrt{2}, -4) \)

- \( P_3\left(\frac{5\pi}{4}\right): x = 3 \sec\frac{5\pi}{4} = 3 \times \left(-\sqrt{2}\right) = -3\sqrt{2} \),
\( y = 4 \tan\frac{5\pi}{4} = 4 \times 1 = 4 \)
So, \( P_3 = (-3\sqrt{2}, 4) \)

- \( P_4\left(\frac{7\pi}{4}\right): x = 3 \sec\frac{7\pi}{4} = 3 \times \sqrt{2} = 3\sqrt{2} \),
\( y = 4 \tan\frac{7\pi}{4} = 4 \times (-1) = -4 \)
So, \( P_4 = (3\sqrt{2}, -4) \)

Step 4: Analyze the quadrilateral formed
The points are:
\( P_1 = (3\sqrt{2}, 4), \quad P_2 = (-3\sqrt{2}, -4), \quad P_3 = (-3\sqrt{2}, 4), \quad P_4 = (3\sqrt{2}, -4) \)
Opposite sides are parallel and equal in length:
- \( P_1P_3 \) and \( P_2P_4 \) are vertical segments of length \( 2 \times 4 = 8 \)
- \( P_1P_4 \) and \( P_2P_3 \) are horizontal segments of length \( 2 \times 3\sqrt{2} = 6\sqrt{2} \)

Step 5: Check if the quadrilateral is a rectangle
Since adjacent sides are perpendicular (one vertical and one horizontal) and lengths are equal for opposite sides, the quadrilateral formed is a rectangle.

Conclusion:
The points \( P_1, P_2, P_3, P_4 \) form a rectangle on the hyperbola.