Question

find the locus of a point P, if the distance of P from the x-axis is always three times its distance from the point (3,- 3)

Answer 1

Let point P is (x, y).

a/c to question,

distance of P from x - axis = 3 × distance of p from the point (3, -3)

use distance formula,

distance of p from x - axis = distance of (x, y) from (x , 0)

[ in case of x-axis , ordinate (y) = 0 ]

= √{(x - x)² + (y - 0)²} = |y|

again, distance between p and (3, -3)

= √{(x - 3)² + (y + 3)² }

so, |y| = 3√{(x - 3)² + (y + 3)²}

squaring both sides

or, y² = 9(x - 3)² + 9(y + 3)²

or, y² = 9x² - 54x + 81 + 9y² + 54y + 81

or, 9x² + 8y² - 54x + 54y + 162 = 0

hence, locus of point p is 9x² + 8y² - 54x + 5y + 162 = 0