Q.Find the equation of the set of all points the sum of whose distances from the points $(3, 0)$ and $(9, 0)$ is 12.

Concept: Locus of a point satisfying a distance-sum condition (definition of an ellipse).

Let $P(x, y)$ be any point on the required locus. The condition states:

The two given points $F_1(3, 0)$ and $F_2(9, 0)$ act as foci. The center of the ellipse is the midpoint: $C = \left(\frac{3+9}{2}, 0\right) = (6, 0)$.

Since the sum of distances equals $2a = 12$, we have $a = 6$. The distance between foci is $2c = |9-3| = 6$, so $c = 3$. …

Understanding the Locus

When a point moves so that the sum of its distances from two fixed points remains constant, it traces an ellipse. The two fixed points are the foci of the ellipse.

Here's why this works: imagine stretching a string between two pins (the foci) and pulling it taut with a pencil. As you move the pencil while keeping the string taut, the total length (sum of distances) stays the same, and you draw an ellipse.

In our problem, the foci are $F_1 = (3, 0)$ and $F_2 = (9, 0)$, and for any point $P = (x, y)$ on the locus, we have:

Finding the Ellipse Equation

1. Identify the center

The center of an ellipse lies at the midpoint of the segment joining the two foci:

2. Find the semi-major axis $a$

By definition of an ellipse, the sum of distances from any point to the two foci equals $2a$:

3. Calculate the focal distance $c$

The distance from the center to each focus is $c$:

(or equivalently, $6 - 3 = 3$)

4. Determine the semi-minor axis $b$

The fundamental relationship for an ellipse is:

Substituting our values: