Question
A point moves such that the sum of its distances from two fixed points and is always . Find (a) the equation of its locus, (b) the eccentricity, (c) the equations of the directrices and (d) the length of the latus rectum.
Solution — Step by Step
Sum of distances from two fixed points = constant — this is the definition of an ellipse with foci , . Sum , so .
The foci are at , so . For an ellipse:
Equation:
Directrices: .
Locus: , , directrices , latus rectum .
Why This Works
Every conic — ellipse, parabola, hyperbola — has two complementary descriptions: a focus-directrix definition and a focus-sum (or focus-difference) definition. The sum-of-distances definition uniquely characterises the ellipse, with equal to the sum.
Once you know and (the foci location), closes the loop. Eccentricity, directrices, and latus rectum are then determined.
Alternative Method
Start from , isolate one root, square twice. After painful algebra, you arrive at , which is the same ellipse. Use this only if asked to derive from definition.
Common Mistake
Students sometimes confuse the formula (ellipse) with (hyperbola). For the ellipse, always — sum of two sides of any triangle exceeds the third side, so the sum exceeds the focal separation . If you’re getting negative , you’ve used the wrong formula.