1. If distance between the points `A(5,3), B(x,-1)` is `5`, then find the value of `x`

Solution:
Points are `A(5,3),B(x,-1)` and distance`=5`

`:. x_1=5,y_1=3,x_2=x,y_2=-1` and `AB=5`

`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=>5=sqrt((x-5)^2+(-1-3)^2)`

`=>25=x^2-10x+25+(16)`

`=>25=x^2-10x+41`

`=>x^2-10x+16 = 0`

`=>x^2-2x-8x+16 = 0`

`=>x(x-2)-8(x-2) = 0`

`=>(x-2)(x-8) = 0`

`=>(x-2) = 0" or "(x-8) = 0`

`=>x = 2" or "x = 8`

Hence value of x are 2 and 8

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2. If distance between the points `A(x,-1), B(3,2)` is `5`, then find the value of `x`

Solution:
Points are `A(x,-1),B(3,2)` and distance`=5`

`:. x_1=x,y_1=-1,x_2=3,y_2=2` and `AB=5`

`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=>5=sqrt((3-x)^2+(2--1)^2)`

`=>25=9-6x+x^2+(9)`

`=>25=x^2-6x+18`

`=>x^2-6x-7 = 0`

`=>x^2+x-7x-7 = 0`

`=>x(x+1)-7(x+1) = 0`

`=>(x+1)(x-7) = 0`

`=>(x+1) = 0" or "(x-7) = 0`

`=>x = -1" or "x = 7`

Hence value of x are -1 and 7

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3. If distance between the points `A(x,2), B(3,-6)` is `10`, then find the value of `x`

Solution:
Points are `A(x,2),B(3,-6)` and distance`=10`

`:. x_1=x,y_1=2,x_2=3,y_2=-6` and `AB=10`

`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=>10=sqrt((3-x)^2+(-6-2)^2)`

`=>100=9-6x+x^2+(64)`

`=>100=x^2-6x+73`

`=>x^2-6x-27 = 0`

`=>x^2+3x-9x-27 = 0`

`=>x(x+3)-9(x+3) = 0`

`=>(x+3)(x-9) = 0`

`=>(x+3) = 0" or "(x-9) = 0`

`=>x = -3" or "x = 9`

Hence value of x are -3 and 9

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4. If distance between the points `A(x,1), B(-1,5)` is `5`, then find the value of `x`

Solution:
Points are `A(x,1),B(-1,5)` and distance`=5`

`:. x_1=x,y_1=1,x_2=-1,y_2=5` and `AB=5`

`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=>5=sqrt((-1-x)^2+(5-1)^2)`

`=>25=1--2x+x^2+(16)`

`=>25=x^2--2x+17`

`=>x^2-2x-8 = 0`

`=>x^2+2x-4x-8 = 0`

`=>x(x+2)-4(x+2) = 0`

`=>(x+2)(x-4) = 0`

`=>(x+2) = 0" or "(x-4) = 0`

`=>x = -2" or "x = 4`

Hence value of x are -2 and 4

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5. If distance between the points `A(x,7), B(1,15)` is `10`, then find the value of `x`

Solution:
Points are `A(x,7),B(1,15)` and distance`=10`

`:. x_1=x,y_1=7,x_2=1,y_2=15` and `AB=10`

`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=>10=sqrt((1-x)^2+(15-7)^2)`

`=>100=1-2x+x^2+(64)`

`=>100=x^2-2x+65`

`=>x^2-2x-35 = 0`

`=>x^2+5x-7x-35 = 0`

`=>x(x+5)-7(x+5) = 0`

`=>(x+5)(x-7) = 0`

`=>(x+5) = 0" or "(x-7) = 0`

`=>x = -5" or "x = 7`

Hence value of x are -5 and 7

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6. If distance between the points `A(1,x), B(-3,5)` is `5`, then find the value of `x`

Solution:
Points are `A(1,x),B(-3,5)` and distance`=5`

`:. x_1=1,y_1=x,x_2=-3,y_1=5` and `AB=5`

`AB=sqrt((x_2-x_1)^2+(y_2 - y_1)^2)`

`=>5=sqrt((-3 - 1)^2+(x-5)^2)`

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

`=>25=x^2-10x+41`

`=>x^2-10x+16 = 0`

`=>x^2-2x-8x+16 = 0`

`=>x(x-2)-8(x-2) = 0`

`=>(x-2)(x-8) = 0`

`=>(x-2) = 0" or "(x-8) = 0`

`=>x = 2" or "x = 8`

Hence value of x are 2 and 8

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7. If distance between the points `A(x,0), B(4,8)` is `10`, then find the value of `x`

Solution:
Points are `A(x,0),B(4,8)` and distance`=10`

`:. x_1=x,y_1=0,x_2=4,y_2=8` and `AB=10`

`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=>10=sqrt((4-x)^2+(8-0)^2)`

`=>100=16-8x+x^2+(64)`

`=>100=x^2-8x+80`

`=>x^2-8x-20 = 0`

`=>x^2+2x-10x-20 = 0`

`=>x(x+2)-10(x+2) = 0`

`=>(x+2)(x-10) = 0`

`=>(x+2) = 0" or "(x-10) = 0`

`=>x = -2" or "x = 10`

Hence value of x are -2 and 10