Question

Find the point on x-axis which is equidistant from the points (2,-2) and (-4,2)

HELP ME​

Answer 1

Let P(m,n) be point on x-axis which is equidistant from the points (2,-2) and (-4,2).

Now, if P(m,n) lies on x-axis then 'n' must be zero.

Therefore , n = 0

If P(m,0) is equidistant from the points (2,-2) and (-4,2) then distance between points P(m,0) and (2,-2) will be equal to the distance between points P(m,0) and (-4,2).

Let d be distance between points P(m,0) and (2,-2) :-

Now using distance formula we will find distance between two given points :-

➝ d² = (2-m)² +(-2-0)²

➝ d² = 4+ m² - 4m +4

➝ d² = 8+ m² - 4m

Let t be distance between points P(m,0) and (-4,2) :-

➝ t² = (-4-m)² +(2-0)²

➝ t² = (-1)²(4+m)² +(2-0)²

➝ t² = 16+m²+8m +4

➝ t² = 20+m²+8m

Since ,P(m,0 ) is equidistant from the points (2,-2) and (-4,2) then d = t and d² = t²

➝ 8+ m² - 4m =20+m²+8m

➝ 8 - 4m =20+8m

➝ 8 - 20 = 8m+4m

➝ -12 = 12m

➝ -12 /12= m

➝ -1 = m

The point on x-axis which is equidistant from the points (2,-2) and (-4,2) = (-1 ,0)

Answer 2

$\huge\bold{Answer:-}$

Given:-

• The points are ( 2 , -2 ) , ( -4 , 2 ) which are equidistant from a point on x axis .

To Find :-

• The point of x axis and the coordinate of it.

Solution:-

Now, As we have to find the point of x axis that means the y is 0

let, the points be a( 2 , -2) , b ( -4 , 2 )

The point from which it is equidistant be O( x , 0 )

so,

As the points are equidistant from each other that means the points have an equal distance from each other from the point O .

AO = OB

NOW,

By using distance formulla we have to find the distance of AO and OB

DISTANCE BETWEEN TWO POINTS

= $\sqrt{(X2 - X1 )² + ( Y2 - Y1)²}$

Applying the above formulla to find the distance of AO AND OB ,

AO = $\sqrt{( x - 2 )² + ( 0 + 2 )² }$

OB = $\sqrt{( x + 4)² + ( 0 -2 )² }$

AO = OB

➱ $\sqrt{( x - 2 )² + ( 0 + 2 )² }$ = $\sqrt{( x + 4)² + ( 0 -2 )² }$

squaring on both sides ,

➱ ($\sqrt{( x - 2 )² + ( 0 + 2 )² }$ )²= ($\sqrt{( x + 4)² + ( 0 -2 )² }$ )²

➱ ( x -2 ) ² + (0 +2)² = ( x + 4)² + ( 0 -2)²

[ applying ( a + b)² = a² + 2ab + b²

and ( a - b) ² = a² - 2ab + b² ]

➱ x² - 4x + 4 + 4 = x ² + 8x + 16 + 4

➱ - 4x - 8x = 16 - 4

➱ -12 x = 12

➱ x = $\dfrac{12}{-12}$

➱ x = -1

so, the point of x axis is -1 .

so, the point of x axis is -1 .And its coordinate will be O( -1 , 0 )