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

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Answer

Answer

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

Answer

Let the given point be A( -2,5) and ( 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 ]
=>(x2−x1)2+(y2−y1)2=(x2−x1)2+(y2−y1)2.
=> [ x - ( -2 ) ]² + ( x - 5 )² = ( x - 2 )² + [ 0 - ( -3 ) ]² .
=> ( 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)

Answer

The point on the x-axis that is equidistant from the points (-2, 5) and (2, -3) can be found by using the distance formula and solving for the x-coordinate. Let's go through the steps to find the coordinates of this point.

Step 1: Understanding the problem
We are given two points, (-2, 5) and (2, -3), and we need to find the point on the x-axis that is equidistant from these two points. This means that the distance from this point to (-2, 5) should be the same as the distance from this point to (2, -3).

Step 2: Finding the distance between the given points
To find the distance between two points, we can use the distance formula, which is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can find the distance between (-2, 5) and (2, -3):

d1 = sqrt((2 - (-2))^2 + (-3 - 5)^2)
= sqrt(4^2 + (-8)^2)
= sqrt(16 + 64)
= sqrt(80)
= 8.94 (approximately)

Step 3: Setting up the equation
Let's assume that the point on the x-axis is (x, 0). We need to find the value of x.

The distance from (x, 0) to (-2, 5) can be found using the distance formula:

d2 = sqrt((x - (-2))^2 + (0 - 5)^2)
= sqrt((x + 2)^2 + 25)

The distance from (x, 0) to (2, -3) can also be found using the distance formula:

d3 = sqrt((x - 2)^2 + (0 - (-3))^2)
= sqrt((x - 2)^2 + 9)

Since the point (x, 0) is equidistant from (-2, 5) and (2, -3), we can set up the equation:

d2 = d3

sqrt((x + 2)^2 + 25) = sqrt((x - 2)^2 + 9)

Step 4: Solving the equation
To solve the equation, we can square both sides:

(x + 2)^2 + 25 = (x - 2)^2 + 9

Expanding and simplifying the equation, we get:

x^2 + 4x + 4 + 25 = x^2 - 4x + 4 + 9

Simplifying further, we have:

4x + 29 = -4x + 13

Adding 4x and subtracting 13 from both sides, we get:

8x = -16

Dividing both sides by 8, we find:

x = -2

Step 5: Final answer
Therefore, the x-coordinate of the point on the x-axis that is equidistant from (-2, 5) and (2, -3) is -2. The y-coordinate is 0 since it lies on the x-axis.

So

Similar Class 10 Doubts

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