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  • If the point P(x,y)  is equidistant from the point A (a-b, a +b ) and B (a-b, a +b ). Prove that bx = ay        &nbsp

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

 

 

 

 
 
 
 
 

Answers (1)

P = (x,y) 

A (a-b, a +b ) and B (a-b, a +b ).

It is given AP = PB 

\sqrt { (x-x_1)^2+( y-y_1)^2} = \sqrt {(x-x_2)^2 + ( y-y_2)^2}\\\\ \sqrt { (x - ( a+b))^2+ ( y - ( b-a ))^2} = \sqrt { x- ( a-b )^2 + ( y - ( a+b ))^2} \\\\

squaring both sides 

\\\\ { (x - ( a+b))^2+ ( y - ( b-a ))^2} = {( x- ( a-b )^2 )+ ( y - ( a+b ))^2} \\\\ x ^2 + ( a+ b )^ 2 - 2 x ( a + b ) + y ^ 2 + ( b - a )^2 - 2 y ( ( b- )) \\\\= x ^ 2 + ( a-b )^2 - 2x ( a-b) + y ^ 2 + ( a+ b)^2 - 2 y ( a+b)

cancelling common terms 

-2 x ( a+b) - 2 y ( b-a ) = - 2 x ( a-b ) - 2 y ( a + b ) \\\\ x ( a + b ) + y ( b -a) = x ( a-b ) + y ( a+ b ) \\\\ ax + bx + by - ay = ax - bx + ay + by \\\\ bx = ay

Hence proved 

 

Posted by

Ravindra Pindel

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