Question

A line $p x + q y + r = 0$ touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Then which of the following statements is true:

• A. $ap^2 + bq^2 = r^2$
• B. $a^2 p^2 + b^2 q^2 = r^2$
• C. $a^2 p + b^2 q = r^2$
• D. $\frac{a^2}{p^2} + \frac{b^2}{q^2} = r$

Hint:

The expression $a^2 p^2 + b^2 q^2 = r^2$ is the standard condition of tangency for the line $px + qy + r = 0$ with respect to the ellipse.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
When a straight line is tangent to a conic section, it must satisfy a specific condition of tangency.
Key Formula or Approach:
The condition for a straight line $y = mx + c$ to touch the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is:
\[ c^2 = a^2 m^2 + b^2 \]

Step 2: Detailed Explanation:
Let us rearrange the given line equation $px + qy + r = 0$ into slope-intercept form $y = mx + c$:
\[ qy = -px - r \]
Assuming $q \neq 0$:
\[ y = \left(-\frac{p}{q}\right) x + \left(-\frac{r}{q}\right) \]
Identify the parameters:
\[ m = -\frac{p}{q} \quad \text{and} \quad c = -\frac{r}{q} \]
Substitute these expressions into the tangency condition:
\[ \left(-\frac{r}{q}\right)^2 = a^2 \left(-\frac{p}{q}\right)^2 + b^2 \]
\[ \frac{r^2}{q^2} = \frac{a^2 p^2}{q^2} + b^2 \]
Multiply the entire equation by $q^2$:
\[ r^2 = a^2 p^2 + b^2 q^2 \]
Rearranging terms:
\[ a^2 p^2 + b^2 q^2 = r^2 \]
Therefore, the statement $a^2 p^2 + b^2 q^2 = r^2$ is true.

Step 3: Final Answer
The correct option is (B).