The normal at a point P on the ellipse having A, A' as vertices and S, S' as foci, bisects an angle?

• A.＜A’PA
• B.＜A’PS
• C.＜S‘PS
• D.＜S'PA

To tackle the question regarding the normal at a point P on an ellipse and its relationship with angles formed by certain points, we first need to clarify some concepts related to ellipses and their properties. The vertices, foci, and the normal line play crucial roles in understanding the geometry of the ellipse.

Understanding the Ellipse

An ellipse is defined as the set of points where the sum of the distances from two fixed points (the foci) is constant. The vertices of the ellipse are the points where it intersects its major axis. In your case, A and A' are the vertices, while S and S' are the foci.

The Normal Line

The normal line at any point P on the ellipse is a line that is perpendicular to the tangent line at that point. This normal line has interesting geometric properties, especially in relation to angles formed with lines drawn to the foci and vertices.

Analyzing the Angles

Now, let’s consider the angles mentioned in your question:

• ∠A'PA: The angle formed between the lines from P to the vertices A and A'.
• ∠A'PS: The angle formed between the lines from P to the vertex A' and the focus S.
• ∠S'PS: The angle formed between the lines from P to the foci S and S'.
• ∠S'PA: The angle formed between the lines from P to the focus S' and the vertex A.

Key Property of the Normal

One of the remarkable properties of the normal line at point P on an ellipse is that it bisects the angle between the line segments drawn to the foci from that point. This means that the normal at P will bisect the angle formed by the lines connecting P to the foci S and S'.

Conclusion on the Angle Bisected

Given this property, the correct answer to your question is that the normal at point P on the ellipse bisects the angle ∠S'PS. Therefore, the answer is C. ∠S'PS.

Visualizing the Concept

To visualize this, imagine standing at point P on the ellipse. If you were to draw lines to the foci S and S', the normal line would split the angle between these two lines evenly. This geometric relationship is a fundamental characteristic of ellipses and is useful in various applications, including optics and astronomy.