Math, asked by sreelakshmiajith, 1 year ago

if point P(x,y) is equidistant from the points A(a+b,b-a) and B(a-b,a+b). prove that bx=ay

Answers

Answered by PARTHtopper
4
Distace between the points (x, y) and  (a+b, b-a) & (a-b, a+b) is equal 

⇒ √{[x - (a + b)]2 + [y - (b -a)]2} = √{x - (a - b)]2 + [y - (a + b)]2}

⇒ x2 + (a + b)2 - 2x(a + b) + y2 + (b - a)2 - 2y(b - a) = x2 + (a - b)2 - 2x(a - b) + y2 + (a + b)2 - 2y(a + b)

⇒ -2ax - 2bx - 2by + 2ay = - 2ax + 2bx - 2ay - 2by
⇒ ay - bx = bx - ay
⇒ 2ay = 2bx
⇒ bx = ay

Hence proved.



PARTHtopper: make me brilliant
Answered by nitthesh7
14
As Point P be equidistant from A and B

Then, AP = BP

So, we know that 

Distance =  √(x₂-x₁)² - (y₂-y₁)²

Let us square on both sides in AP = BP 

AP² = BP²

And so the roots get cancelled

Then, 

(x-a-b)² + (y+a-b)² = (x-a+b)² + (y-a-b)²

x² + a² + b² - 2ax + 2ab - 2bx + y² + a² + b² + 2ay - 2ab -2by
                   
               = x² + a² + b² -2ax + 2bx - 2ab + y² + a² + b² - 2ay + 2ab - 2by

(From Formula (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ac )

2ay - 2bx = 2bx - 2ay

(cancelling the opp sides therms as it becomes opp signs)

2ay + 2ay = 2bx + 2bx

4ay = 4bx

ay = bx

(cancelling 4 on both sides)

bx = ay


HENCE PROVED



☺ Hope this Helps 
 

nitthesh7: tq for brainliest
sreelakshmiajith: tq for the answer, really helped me
nitthesh7: ☺☺☺
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