Find Eccentricity of Ellipse Orthogonal to Parabola at Latus Rectum

To find the eccentricity of the ellipse, we follow these steps:

Step 1: Determine the coordinates of point $P$

The given parabola is $y^2 = 4\lambda x$. The focus of this parabola is at $(\lambda, 0)$, and the end points of its latus rectum are $(\lambda, 2\lambda)$ and $(\lambda, -2\lambda)$.

Without loss of generality, let $P = (\lambda, 2\lambda)$, where $\lambda > 0$.

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Step 2: Find the slope of the tangent to the parabola at point $P$

Differentiating the equation of the parabola $y^2 = 4\lambda x$ with respect to $x$: $2y \frac{dy}{dx} = 4\lambda \implies \frac{dy}{dx} = \frac{2\lambda}{y}$

Evaluating the derivative at point $P(\lambda, 2\lambda)$, we get the slope $m_1$ of the tangent to the parabola: $m_1 = \left. \frac{dy}{dx} \right|_{(\lambda, 2\lambda)} = \frac{2\lambda}{2\lambda} = 1$

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Step 3: Find the slope of the tangent to the ellipse at point $P$

The equation of the ellipse is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Differentiating implicitly with respect to $x$: $\frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{b^2 x}{a^2 y}$

Evaluating the derivative at point $P(\lambda, 2\lambda)$, we get the slope $m_2$ of the tangent to the ellipse: $m_2 = \left. \frac{dy}{dx} \right|_{(\lambda, 2\lambda)} = -\frac{b^2 \lambda}{a^2 (2\lambda)} = -\frac{b^2}{2a^2}$

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Step 4: Use the orthogonality condition

Since the tangents to the parabola and the ellipse at point $P$ are perpendicular to each other, their slopes must satisfy: $m_1 \cdot m_2 = -1$

Substitute the values of $m_1$ and $m_2$: $1 \cdot \left(-\frac{b^2}{2a^2}\right) = -1$ $\frac{b^2}{2a^2} = 1 \implies b^2 = 2a^2$

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Step 5: Calculate the eccentricity of the ellipse

Since $b^2 = 2a^2 > a^2$, the major axis of the ellipse lies along the y-axis ($b > a$).

The eccentricity $e$ of such an ellipse is given by: $e = \sqrt{1 - \frac{a^2}{b^2}}$

Substituting $b^2 = 2a^2$, we get: $e = \sqrt{1 - \frac{a^2}{2a^2}} = \sqrt{1 - \frac{1}{2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$

Thus, the correct option is (A).