P(x,y) moves such that its distance from point Q(4,8)is twiceits distance from point P(-2,3). Find the equation of the locus of moving point P.
?((x-4)^2+(y-8)^2) = 2?((x+2)^2+(y-3)^2)
square both sides and collect terms. It will be a circle.
http://www.wolframalpha.com/input/?i=circle+((x-4)%5E2%2B(y-8)%5E2)+%3D+4((x%2B2)%5E2%2B(y-3)%5E2)
To find the equation of the locus of the moving point P, let's first analyze the given conditions:
1. The distance of point P from point Q is twice its distance from point P.
2. Point Q is fixed at coordinates (4,8).
3. Point P is the moving point with unknown coordinates (x,y).
Let's calculate the distances between P, Q, and the conditions using the distance formula:
The distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Applying this formula, we can determine the distances and form the equation based on the given conditions.
Distance between P and Q:
dPQ = sqrt((x - 4)^2 + (y - 8)^2)
Distance between P and P:
dPP = sqrt((x - (-2))^2 + (y - 3)^2) = sqrt((x + 2)^2 + (y - 3)^2)
Now, we have the condition that the distance of P from Q is twice its distance from P:
dPQ = 2 * dPP
Substituting the equations for dPQ and dPP, we get:
sqrt((x - 4)^2 + (y - 8)^2) = 2 * sqrt((x + 2)^2 + (y - 3)^2)
To remove the square roots, we can square both sides of the equation:
(x - 4)^2 + (y - 8)^2 = 4 * ((x + 2)^2 + (y - 3)^2)
Expanding the equation, we have:
x^2 - 8x + 16 + y^2 - 16y + 64 = 4x^2 + 16x + 16 + 4y^2 - 24y + 36
Simplifying further:
3x^2 + 12x + 3y^2 - 8y - 24 = 0
Thus, the equation of the locus of the moving point P is:
3x^2 + 3y^2 + 12x - 8y - 24 = 0