If the tangent to the ellipse `x^(2)/a^(2) + y^(2)/b^(2) = 1` makes intercepts p and q on the coordinate axes, then `a^(2)/p^(2) + b^(2)/q^(2) = `

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the value of \( \frac{a^2}{p^2} + \frac{b^2}{q^2} \) where \( p \) and \( q \) are the x-intercept and y-intercept of the tangent to the ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

### Step-by-Step Solution:

1. **Equation of the Ellipse**:
The equation of the ellipse is given as:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

2. **Equation of the Tangent**:
The equation of the tangent to the ellipse in parametric form is:
\[ \frac{bx \cos \theta}{a} + \frac{ay \sin \theta}{b} = 1 \]
where \( \theta \) is the angle at which the tangent touches the ellipse.

3. **Finding the x-intercept (p)**:
To find the x-intercept, set \( y = 0 \) in the tangent equation:
\[ \frac{bx \cos \theta}{a} = 1 \]
Solving for \( x \):
\[ x = \frac{a}{b \cos \theta} \]
Thus, the x-intercept \( p \) is:
\[ p = \frac{a}{\cos \theta} \]

4. **Finding the y-intercept (q)**:
To find the y-intercept, set \( x = 0 \) in the tangent equation:
\[ \frac{ay \sin \theta}{b} = 1 \]
Solving for \( y \):
\[ y = \frac{b}{a \sin \theta} \]
Thus, the y-intercept \( q \) is:
\[ q = \frac{b}{\sin \theta} \]

5. **Substituting p and q into the required expression**:
Now we substitute \( p \) and \( q \) into the expression \( \frac{a^2}{p^2} + \frac{b^2}{q^2} \):
\[ \frac{a^2}{p^2} = \frac{a^2}{\left(\frac{a}{\cos \theta}\right)^2} = \frac{a^2 \cos^2 \theta}{a^2} = \cos^2 \theta \]
\[ \frac{b^2}{q^2} = \frac{b^2}{\left(\frac{b}{\sin \theta}\right)^2} = \frac{b^2 \sin^2 \theta}{b^2} = \sin^2 \theta \]

6. **Combining the results**:
Now we can combine the results:
\[ \frac{a^2}{p^2} + \frac{b^2}{q^2} = \cos^2 \theta + \sin^2 \theta \]

7. **Using the Pythagorean Identity**:
We know from trigonometry that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

8. **Final Result**:
Therefore, we conclude that:
\[ \frac{a^2}{p^2} + \frac{b^2}{q^2} = 1 \]

### Final Answer:
\[ \frac{a^2}{p^2} + \frac{b^2}{q^2} = 1 \]