Two equal ellipse of eccentercity e is placed with their axes right angle two each other and have a common focus S. If PQ is a common tangent to both the ellipse find the angle PSQ.

To tackle this problem involving two equal ellipses with a shared focus, we need to analyze the configuration and apply some geometric principles. First, let's visualize the setup: we have two ellipses oriented perpendicularly, both sharing a focus point S. The tangents to these ellipses will intersect at some angle, which is what we’re trying to determine.

Understanding Ellipses and Their Properties

An ellipse is defined by its semi-major axis (a), semi-minor axis (b), and eccentricity (e), where the relationship between these elements is given by the equation:

• e = √(1 - (b²/a²))

Since the two ellipses are equal, they share the same eccentricity, which influences their shape and focus. The common focus S is a critical point as it provides a reference for the tangents.

Positioning the Ellipses

Let’s denote the first ellipse as E1 with its center at the origin (0,0) and its axes aligned along the x and y axes. The second ellipse, E2, will be rotated 90 degrees. The foci of both ellipses will be at coordinates determined by the eccentricity. For both ellipses, the focus S will lie at a distance of ae from the center along their respective axes.

Finding the Tangent Line PQ

The tangent line PQ that touches both ellipses can be characterized by its geometric properties. Since the ellipses are symmetric and equal, the tangent will maintain this symmetry. The angle we want to find, angle PSQ, can be analyzed using the properties of tangents to ellipses.

Calculating the Angle PSQ

When a tangent touches an ellipse, the angle that it makes with the line segment connecting the focus to the point of tangency is equal for both ellipses because of their equal eccentricity and symmetric properties. Let’s denote the angle between the line connecting S to the point P on E1 and the line PQ as θ. The angle PSQ will therefore be twice this angle due to the tangents to both ellipses:

• Angle PSQ = 2θ

As the ellipses are equal and oriented perpendicularly, the angle θ can be derived from the geometry of the ellipse. For a common tangent PQ, the angle PSQ can be shown to be:

• Angle PSQ = 90 degrees

This conclusion arises because the two tangential lines at the point of tangency effectively create right angles to the line joining the focus to the tangential point, maintaining the perpendicular relationship between the axes of the ellipses and the tangent line.

Final Thoughts

In summary, the angle PSQ formed by the common tangent to the two ellipses is a direct result of their symmetrical properties and geometric configuration. Understanding the relationship between the foci and the tangents allows us to see that the angle is consistently 90 degrees due to the orthogonality of the axes of the ellipses. This elegant interplay of geometry and symmetry is one of the fascinating aspects of studying conic sections!