Question

Find the coordinates of the point on x-axis which is equidistant from the points ( - 2, 5 ) and ( 2, - 3 ).

Answer 1

Hey there !!

Let the given points be A( -2 , 5 ) and B( 2 , -3 ) .

And, let the required point on x-axis be P( x , 0 ) .

Then,

→ PA = PB . [ A and B are equidistant from P ]

[ Squaring both side ]

=> PA² = PB² .

[ Using distance formula ]

$= > {(x_2 - x_1)}^{2} + {(y_2 - y_1)}^{2} = {(x_2 - x_1)}^{2} + {(y_2 - y_1)}^{2} .$

=> [ x - ( -2 ) ]² + ( x - 5 )² = ( x - 2 )² + [ 0 - ( -3 ) ]² .

=> ( x + 2 )² + ( 0 - 5 )² = ( x - 2 )² + ( 0 + 3 )² .

=> ( x + 2 )² + 25 = ( x - 2 )² + 9 .

=> ( x + 2 )² - ( x - 2 )² = 9 - 25 .

=> [ ( x + 2 ) + ( x - 2 ) ] [ ( x + 2 ) - ( x - 2 ) ] = - 16 .

=> ( x + 2 + x - 2 ) ( x + 2 - x + 2 ) = - 16 .

=> ( 2x ) ( 4 ) = - 16 .

=> 8x = - 16 .

=> x = -16/8 .

•°• x = -2 .

✔✔ Hence, the required point on x-axis is ( -2 , 0 ) ✅✅ .

THANKS

#BeBrainly.

Answer 2

Answer:

the coordinates of the point on x-axis which is equidistant from the points ( - 2, 5 ) and ( 2, - 3 ) are ( - 2 , 0 ) .

Step-by-step explanation:

Given that the point which is equidistant from ( - 2 , 5 ) and ( 2 , - 3 ) lies on x - axis, so y- coordinate of that point should be 0.

Let the required point be P( p , 0 ).

Note : Section formula can't be applied, according to section formula P is the mid point of given points but ( - 3 + 5 ) / 2 is not equal to 0. It means that the required point is not in the straight line( which is joining the given points ).

By Distance formula

$\boxed{ \bold{Distance=\sqrt{(x_{2}-x_{1} )^2 + ( y_{2} - y_{1} )^2}}}$

⇒ Distance between ( - 2 , 5 ) and ( p , 0 ) = distance between ( 2 , - 3 ) and ( p , 0 ).

$\implies \sqrt{(0-5)^2+\{p-(-2)\})^2}= \sqrt{\{0-(-3)\}^2 + ( p -2 )^2}\\\\\implies ( - 5 )^2 + ( p + 2 )^2 = 3^2 + ( p -2 )^2 \\\\\implies 25 - 9 = ( p - 2 )^2 - ( p + 2 )^2\\\\\implies 16 = ( p - 2 - p - 2 )( p - 2+p+2)\\\\\implies 16 = -4( 2p ) \\\\\implies 16= -8p\\\\\implies -\dfrac{16}{8} =p\\\\\implies - 2 = p$

Therefore, the point which is equidistant from the given points is ( - 2 , 0 ).