Question

If R is any point on the x-axis and Q is any point on the y-axis and P is a variable point on RQ with RP = b, PQ = a. then find the equation of locus of P

Answer 1

P = (x, 0)

Q = (0, y)

R (h, k) be a point on RQ such that PR : RQ = b : a

∴ P = `(("a"x)/("a" + "b"), ("b"y)/("a" + "b"))`

⇒ `(("a"x)/("a" + "b"), ("b"y)/("a" + "b"))` = (h, k)

⇒ ax = (a + b)h

⇒ x = `("a" + "b")/"a" "h"`

by = (a + b)k

⇒ y = `("a" + "b")/"b" "k"`

From the right angled triangle OQR, OR² + OQ² = QR² 

(i.e) X² + Y² = (a + b)²

⇒ `[("a" + "b")/"a" "h"]^2 + [("a" + "b")/"b" "k"]^2` = (a + b)²

⇒ `("a" + "b")^2/"a"^2 "h"^2 + ("a" + "b")^2/"b"^2 "k"^2` = (a + b)²

÷ by (a + b)², `"h"^2/"a"^2 + "k"2/"b"^2` = 1

Locus is `x^2/"a"^2 + y^2/"b"^2` = 1