Question

What curve does the locus represent if the locus of a moving point which is always equidistant from the points (2, -1) and (3, 2)?

(a) Hyperbola

(b) Straight line

(c) Ellipse

(d) Circle

I have been asked this question in an international level competition.

My question is based upon Loci of Points topic in section Lettering Practice, Scales & Curves of Civil Engineering Drawing

Answer 1

The correct choice is (b) Straight line

To elaborate: Let A (2, -1) and B (3, 2) be the given points and (x, y) be the co-ordinates of a point P on the required locus. Then,

PA^2 = (x – 2)^2 + (y + 1)^2 and PB^2 = (x – 3)^2 + (y – 2)^2

By problem, PA = PB or, PA^2 = PB^2

or, (x – 2)^2 + (y + 1)^2 = (x – 3)^2 + (y – 2)^2  or, x^2 – 4x + 4 + y^2 + 2y + 1 = x^2 – 6x + 9 + y^2 – 4y + 4

or, 2x + 6y = 8 or, x + 3y = 4 ……… (1)

Clearly, equation (1) is a first degree equation in x and y; hence, the locus of P is a straight line whose equation is x + 3y = 4.