Question

A(2,3)&B(-3,4)are two given points.find the equation of locus of p so that the area of the triangle pab is 8.5

Answer 1

Let point P is (x , y).

a/c to question,

area of ∆PAB = 8.5 , where A(2, 3) and B(-3,4) .

we know, if three points A(x1, y1), B(x2,y2) and C(x3,y3) are joined to form a triangle ABC.

then, area of triangle ∆ABC = 1/2|x1 (y2-y3) + x2(y3-y1) + x3(y1-y2)|

here given, points ; P(x, y), A (2, 3) and B(-3, 4) are joined to form triangle ∆PAB.

so, area of ∆PAB = 1/2 | x(3 - 4) + 2(4 - y) + (-3)(y - 3) |

or, 8.5 = 1/2 |-x + 8 - 2y - 3y + 9 |

or, 17 = |-x - 5y + 17|

or, ± 17 = -x - 5y + 17

or, - x - 5y + 17 = 17 => x + 5y = 0

and -x - 5y + 17 = -17 => x + 5y - 34 = 0

hence, there are two equations ; x + 5y = 0 and x + 5y - 34 = 0. These are required locus of point P.