A and B are two points whose co-ordinates are [-5, 3] and [2, 4], respectively. A point P moves in such a manner that the ratio PA
B = 3:2. Find the equation of the locus traced out by P. What curve does it represent?
The solution is given as:
Let [x, y] be the co-ordinate for any position P. Then 4[PA]<sup>2</sup> = 9[PB]<sup>2</sup>. Applying the Distance Formula, we get the following:
. . .4[(x + 5)<sup>2</sup> + (y - 3)<sup>2</sup>] = 9[(x - 2)<sup>2</sup> + (y - 4)<sup>2</sup>]
Solving this, we get:
. . .x<sup>2</sup> + y<sup>2</sup> - [76x]/5 - [48y]/5 + [44]/5 = 0
. . .sqrt[(x - 38/5)<sup>2</sup> + (y - 24/5)<sup>2</sup>] = 6sqrt[2]
This shows that the distance of the moving point [x, y] from the fixed point [38/5, 24/5] is constant, namely 6sqrt[2]. This identifies the locus to be a circle.
My question:
How do we get that [x, y] is a moving point, when we have used a distance ormula by way of which point P is only intermidiate between A & B along a straight line?
Regards,
Sujoy
B = 3:2. Find the equation of the locus traced out by P. What curve does it represent?The solution is given as:
Let [x, y] be the co-ordinate for any position P. Then 4[PA]<sup>2</sup> = 9[PB]<sup>2</sup>. Applying the Distance Formula, we get the following:
. . .4[(x + 5)<sup>2</sup> + (y - 3)<sup>2</sup>] = 9[(x - 2)<sup>2</sup> + (y - 4)<sup>2</sup>]
Solving this, we get:
. . .x<sup>2</sup> + y<sup>2</sup> - [76x]/5 - [48y]/5 + [44]/5 = 0
. . .sqrt[(x - 38/5)<sup>2</sup> + (y - 24/5)<sup>2</sup>] = 6sqrt[2]
This shows that the distance of the moving point [x, y] from the fixed point [38/5, 24/5] is constant, namely 6sqrt[2]. This identifies the locus to be a circle.
My question:
How do we get that [x, y] is a moving point, when we have used a distance ormula by way of which point P is only intermidiate between A & B along a straight line?
Regards,
Sujoy
