The sum of the focal distances of a point on an ellipse is constant and equal to the
length of major axis
Understanding the Sum of Focal Distances on an Ellipse
The question asks about a fundamental property of an ellipse: the sum of the focal distances of any point lying on the ellipse. This property is one of the defining characteristics of an ellipse.
Let's consider an ellipse with its two foci, denoted as \( F_1 \) and \( F_2 \). For any point \( P \) that lies on the ellipse, the sum of the distances from \( P \) to the two foci is always constant. Mathematically, this can be written as:
\( PF_1 + PF_2 = \text{constant} \)
This constant sum is equal to the length of the major axis of the ellipse.
What is the Major Axis?
The major axis is the longest diameter of the ellipse. It passes through both foci and the center of the ellipse. If the length of the semi-major axis is denoted by \( a \), then the length of the major axis is \( 2a \).
Relating Focal Distances to the Major Axis
Consider the points where the ellipse intersects the major axis. Let these points be \( V_1 \) and \( V_2 \) (the vertices). These points are on the ellipse. By the definition of the ellipse, the sum of the distances from \( V_1 \) to the foci must be constant, and similarly for \( V_2 \).
For point \( V_1 \): \( V_1F_1 + V_1F_2 \)
For point \( V_2 \): \( V_2F_1 + V_2F_2 \)
Due to the symmetry of the ellipse, the sum \( V_1F_1 + V_1F_2 \) and \( V_2F_1 + V_2F_2 \) are equal to the length of the major axis \( V_1V_2 \). Thus, for any point \( P \) on the ellipse, the sum of its distances from the foci \( PF_1 + PF_2 \) is equal to the length of the major axis.
Analyzing the Options
Option 1: length of minor axis. The minor axis is the shortest diameter, perpendicular to the major axis. Its length is \( 2b \), where \( b \) is the semi-minor axis length. This is not equal to the constant sum of focal distances.
Option 2: length of major axis. This is the length \( 2a \), which is indeed equal to the constant sum of the focal distances \( PF_1 + PF_2 \) for any point \( P \) on the ellipse.
Option 3: length of latus rectum. The latus rectum is a chord through a focus perpendicular to the major axis. Its length is \( \frac{2b^2}{a} \). This is not equal to the constant sum of focal distances.
Option 4: sum of the lengths of semi-major and semi-minor axes. This is \( a+b \). This sum is not equal to the constant sum of focal distances (\( 2a \)).
Therefore, the sum of the focal distances of a point on an ellipse is constant and equal to the length of the major axis.
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