Question

If (3,2) and (-3,2) are two vertices of equilateral triangle which conrains the origin, find the third vertex. If point P(x,y) is equidistant from the points A(a+b,b-a) and B(a-b,a+b), then prove that bx=ay

Answer 1

1). given,
A(3,2) and B(-3,2) are two vertices of equilateral triangle. in equilateral triangle all sides are equal.
distance between A and B =√(3+3)²+(2-2)²
=√6²+0²=√6²=6
Distance between A and C=√(3-x)²+(2-y)²
6=√(3-x)²+(2-y)²
Squaring on both sides.
6²=(3-x)²+(2-y)²
36=3²+x²-6x+2²+y²-4y
36=9+x²-6x+4+y²-4y
23=x²+y²-6x-4y...........................(1)
Distance between B and C=√(-3-x)²+(2-y)²
6=√(-3-x)²+(2-y)²
squaring on both sides
6²=(-3-x)²+(2-y)²
36=9+x²+6x+4+y²-4y
23=x²+y²+6x-4y........................(2)
from (1) and(2)
x²+y²-6x-4y=x²+y²+6x-4y
-6x-6x=0
-12x=0
x=0
Substitute x=0 in (1)
y²-4y=23
y²-4y-23=0
y=-(-4)±√(-4)²-4(1)(-23)/2
y=4±√16+92/2
y=4±√108/2
y=4±(6√3)/2
y=(4±6√3)/2
y=4+6√3/2. , y=4-6√3/2
y=2+2√3, y=2-2√3

(x,y)=(0,2±2√3)


2). p(x,y) is the midpoint of a and b
a(a+b,b-a) b(a-b,a+b)

(x,y)=(a+b+a-b)/2 , (b-a+a+b)/2
(x,y)=(2a/2,2b/2)
x=a ...................(1). and y=b=>b=y................(2)
multiply (1) and (2)
bx=ay
Hence proved

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