A point P moves so as to satisfy the ratio OP:PA=1:3,where O is the origin and A is the point (-4,0).Find the equation of the locus of P.
OP /PA = 1/3
3OP = AP
now follow the same method I just showed you in your previous post
To find the equation of the locus of point P, we need to determine the relationship between the coordinates of P (x,y) that satisfy the given ratio.
Let's start by representing point P as (x, y). The distance between points O(0,0) and A(-4,0) can be found using the distance formula as follows:
OP = √((x - 0)^2 + (y - 0)^2) = √(x^2 + y^2)
Similarly, the distance between points A(-4,0) and P(x, y) is given by:
PA = √((x - (-4))^2 + (y - 0)^2) = √((x + 4)^2 + y^2)
According to the given ratio OP:PA = 1:3, we have:
OP / PA = 1/3
Substituting the obtained values of OP and PA, we get:
√(x^2 + y^2) / √((x + 4)^2 + y^2) = 1/3
Now, to eliminate the square roots, we can square both sides of the equation:
(x^2 + y^2) / ((x + 4)^2 + y^2) = (1/3)^2
Simplifying further, we have:
3(x^2 + y^2) = (x + 4)^2 + y^2
Expanding and rearranging terms, we get:
3x^2 + 3y^2 = x^2 + 8x + 16 + y^2
Combining like terms, we have:
2x^2 + 2y^2 - 8x - 16 = 0
Dividing by 2, we obtain the final equation of the locus of point P:
x^2 + y^2 - 4x - 8 = 0
Hence, the equation of the locus of point P is x^2 + y^2 - 4x - 8 = 0.