+105 Write a simplified equation for the set of points that are exactly twice as far from (12,0) as they are from (0,0).
+4 A point (x,y)(x, y)(x,y) is twice as far from (12,0) as from (0,0).
Step 1: Write distances
From (0,0)(0,0)(0,0):
x2+y2\sqrt{x^2 + y^2}x2+y2
From (12,0)(12,0)(12,0):
(x−12)2+y2\sqrt{(x-12)^2 + y^2}(x−12)2+y2
Step 2: Apply condition
(x−12)2+y2=2x2+y2\sqrt{(x-12)^2 + y^2} = 2\sqrt{x^2 + y^2}(x−12)2+y2=2x2+y2
Step 3: Square both sides
(x−12)2+y2=4(x2+y2)(x-12)^2 + y^2 = 4(x^2 + y^2)(x−12)2+y2=4(x2+y2)
Step 4: Expand
x2−24x+144+y2=4x2+4y2x^2 - 24x + 144 + y^2 = 4x^2 + 4y^2x2−24x+144+y2=4x2+4y2
Step 5: Simplify
−3x2−3y2−24x+144=0-3x^2 - 3y^2 - 24x + 144 = 0−3x2−3y2−24x+144=0
Divide by −3-3−3:
x2+y2+8x−48=0x^2 + y^2 + 8x - 48 = 0x2+y2+8x−48=0
Final Answer:
👉 x2+y2+8x−48=0x^2 + y^2 + 8x - 48 = 0x2+y2+8x−48=0
+130577 2 sqrt (x^2 + y^2)) = sqrt [ (x-12)^2 + y^2]
4 [ x^2 + y^2] = [ x^2 - 24x + 144 + y^2]
4x^2 + 4y^2 = x^2 - 24x + 144 + y^2
3x^2 + 3y^2 = 144 - 24x
x^2 + y^2 = 48 - 8x
x^2 + 8x + y^2 = 48 ........complete the square on x
x^2 + 8x + 16 + y^2 = 48 + 16
(x + 4)^2 + y^2 = 64
The set of points lie on a circle with center (-4,0) and radius = 8
