1. Find the point on the x-axis which is equidistant from `A(5,4)` and `B(-2,3)`

Solution:
Let `P(x,0)` be the point on X-axis, which is the equidistance form `A(5,4)` and `B(-2,3)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(x-5)^2+(0-4)^2=(x+2)^2+(0-3)^2`

`=>x^2-10x+25+16=x^2 +4x+4+9`

`=>-10x-4x=4+9-25-16`

`=>-14x=-28`

`=>x=2`

Thus, the required point is `P(2,0)`.

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2. Find the point on the x-axis which is equidistant from `A(2,-5)` and `B(-2,9)`

Solution:
Let `P(x,0)` be the point on X-axis, which is the equidistance form `A(2,-5)` and `B(-2,9)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(x-2)^2+(0+5)^2=(x+2)^2+(0-9)^2`

`=>x^2-4x+4+25=x^2 +4x+4+81`

`=>-4x-4x=4+81-4-25`

`=>-8x=56`

`=>x=-7`

Thus, the required point is `P(-7,0)`.

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3. Find the point on the x-axis which is equidistant from `A(7,6)` and `B(-3,4)`

Solution:
Let `P(x,0)` be the point on X-axis, which is the equidistance form `A(7,6)` and `B(-3,4)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(x-7)^2+(0-6)^2=(x+3)^2+(0-4)^2`

`=>x^2-14x+49+36=x^2 +6x+9+16`

`=>-14x-6x=9+16-49-36`

`=>-20x=-60`

`=>x=3`

Thus, the required point is `P(3,0)`.

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4. Find the point on the x-axis which is equidistant from `A(-2,5)` and `B(2,-3)`

Solution:
Let `P(x,0)` be the point on X-axis, which is the equidistance form `A(-2,5)` and `B(2,-3)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(x+2)^2+(0-5)^2=(x-2)^2+(0+3)^2`

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

`=>4x+4x=4+9-4-25`

`=>8x=-16`

`=>x=-2`

Thus, the required point is `P(-2,0)`.

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5. Find the point on the x-axis which is equidistant from `A(5,-2)` and `B(-3,2)`

Solution:
Let `P(x,0)` be the point on X-axis, which is the equidistance form `A(5,-2)` and `B(-3,2)`.

`:. PA=PB`

`=>PA^2=PB^2`

`=>(x-5)^2+(0+2)^2=(x+3)^2+(0-2)^2`

`=>x^2-10x+25+4=x^2 +6x+9+4`

`=>-10x-6x=9+4-25-4`

`=>-16x=-16`

`=>x=1`

Thus, the required point is `P(1,0)`.