# Locus and its equations of co-ordinates

Definition of Locus: The curve described by a point which moves under given condition or conditions is called its locus.

Equation to the locus of a point: The equation to the locus of a point is the relation which is satisfied by the coordinates of every point on the locus of the point.

Algorithm to find the locus of a point: Assume the coordinates of the point (h, k) whose locus to be found.

1. Write the given condition in mathematical form involving h, k.
2. Eliminate the variables if any
3. Replace h by x and k by y in the result (3).

SOME USEFUL POINTS:

A. In order to prove that a given figure is a

• Square, prove that the four sides are equal and the diagonals are also equal.
• Rhombus, prove that the four sides are equal.
• Rectangle, prove that opposite sides are equal and the diagonals are also equal.
• A parallelogram, prove that the opposite sides are equal.
• Parallelogram but not a rectangle, prove that its opposite sides are equal but the diagonals are not equal.
• A rhombus but not a square, prove that it’s all sides are equal but the diagonals are not equal.

B. For three points to be collinear, prove that the sum of the distance between two pairs of points, is equal to the third pair of points.

1. Find the locus of the point which is equidistant to the coordinate axes.

Sol: Let P(x₁, y₁) be a point in the locus

The distance from P to x – axis is |y₁|

The distance from P to y – axis is |x₁|

Given condition is |y₁| = |x₁| ⇒ y²₁ = x²₁ ⇒ x²₁ – y²₁ = 0

The equation to the locus of P is x² – y² = 0

2. The ends of the hypotenuse of a right-angled triangle are (0, 6), (6, 0). Find the locus of the third vertex?

Sol: Let A (0, 6), B (6, 0) be the ends of the hypotenuse and P (x, y) be the third vertex.Given condition is ∠APB = 90⁰

⇒ PA² + PB² = (AB)²

⇒ x² + (y – 6) ² + (x – 6) ² + y² = 6² + 6²

⇒ x² + y²

⇒ x² + y² – 12y + 36 + x² – 12x + 36 – y² = 72

⇒ 2x² + 2y² – 12x – 12y = 0

⇒ x² + y² – 6x – 6y = 0

∴ The locus of P is x² + y² – 6x – 6y = 0