Understanding Ellipse Properties: Foci and Point Distances

The question asks us to find the sum of the distances from any point P on a given ellipse to its two foci, Q and R. This relates to a fundamental definition of an ellipse.

Definition of an Ellipse

An ellipse is defined as the set of all points P in a plane such that the sum of the distances from P to two fixed points (called the foci, Q and R) is a constant value. This constant sum is always equal to the length of the major axis of the ellipse.

Mathematically, for any point P on the ellipse, the property states:

\( PQ + PR = 2a \)

where '\(2a\)' is the length of the major axis.

Analyzing the Ellipse Equation

The equation of the ellipse provided is:

\( 4x^2 + 9y^2 = 1 \)

To find the length of the major axis (\(2a\)), we first need to convert this equation into the standard form of an ellipse equation. The standard forms are:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (if the major axis is horizontal)
• \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) (if the major axis is vertical)

Divide the given equation by 1 to get the standard form:

\( \frac{4x^2}{1} + \frac{9y^2}{1} = 1 \)

Rewrite this as:

\( \frac{x^2}{1/4} + \frac{y^2}{1/9} = 1 \)

Identifying 'a' and Calculating the Major Axis Length

Now, compare this equation with the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

• We have \(a^2 = 1/4\) and \(b^2 = 1/9\).
• Since \(1/4 > 1/9\), \(a^2\) is indeed under the \(x^2\) term, confirming the major axis is horizontal.
• The length of the semi-major axis, \(a\), is \(a = \sqrt{1/4} = 1/2\).
• The length of the major axis, \(2a\), is \(2a = 2 \times (1/2) = 1\).

Conclusion for PQ + PR

Based on the definition of the ellipse, the sum of the distances from any point P on the ellipse to the foci Q and R is equal to the length of the major axis (\(2a\)).

Therefore, \( PQ + PR = 2a = 1 \).