Question

The X and Y intercepts of the tangent to the hyperbola \(\frac{x^2}{20} - \frac{y^2}{5} = 1\) which is perpendicular to the line \(4x + 3y = 7\), are respectively

• A. \(\frac{-10}{3}, \frac{-5}{3}\)
• B. \(\frac{10}{3}, \frac{-5}{2}\)
• C. \(\frac{10}{3}, \frac{5}{2}\)
• D. \(\frac{10}{3}, \frac{5}{3}\)

Hint:

In conic questions, “intercepts” in MCQs often mean intercept lengths, which are taken positive.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Two lines are perpendicular if the product of their slopes is -1. We first find the slope of the tangent and then its equation.
Step 2: Key Formula or Approach:
1. Slope of tangent \(m = -\frac{1}{\text{slope of line}}\).
2. Equation of tangent to hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is \(y = mx \pm \sqrt{a^2m^2 - b^2}\).
Step 3: Detailed Explanation:
The given line is \(4x + 3y = 7\). Its slope \(m_1 = -4/3\).
Slope of the required tangent \(m = \frac{-1}{m_1} = 3/4\).
Hyperbola parameters: \(a^2 = 20, b^2 = 5\).
Tangent equation:
\[ y = \frac{3}{4}x \pm \sqrt{20 \cdot (\frac{3}{4})^2 - 5} \] \[ y = \frac{3}{4}x \pm \sqrt{20 \cdot \frac{9}{16} - 5} = \frac{3}{4}x \pm \sqrt{\frac{45}{4} - \frac{20}{4}} \] \[ y = \frac{3}{4}x \pm \sqrt{\frac{25}{4}} = \frac{3}{4}x \pm \frac{5}{2} \] Let's take \(y = \frac{3}{4}x - \frac{5}{2}\).
X-intercept (put \(y=0\)): \(0 = \frac{3}{4}x - \frac{5}{2} \implies 3x = 10 \implies x = 10/3\).
Y-intercept (put \(x=0\)): \(y = -5/2\).
The intercepts are \(10/3\) and \(-5/2\), which matches option (B).
Step 4: Final Answer:
The intercepts are \(\frac{10}{3}\) and \(\frac{-5}{2}\).