Find the number of distinct normals that can be drawn to the ellipse $\frac{x^2}{69} + \frac{y^2}{25} = 1$ from the point $P(0, 6)$.
Step-by-Step Solution
Key Concept: For a given point, multiple normals to an ellipse can pass through it, corresponding to different values of the parameter $\theta$.
The normal to the ellipse is given by $13\sec\theta - 5\cos\theta = 144$. If the normal passes through $(0, 6)$, then $\sin\theta = -\frac{5}{24}$. Since $\cos\theta$ has two values for this $\sin\theta$, there are two possible normals. One of these normals is the $y$-axis, while the other is a different line.
Correct Answer: 3