Question

Determine the value of x when (-1,-1)is equidistant from (0,2)and(x,-4)?

Answer 1

`x=0 or -2`

Explanation:

Using distance formula / Pythagorean theorem,

Let the distance between `(-1,-1)` and `(0,2)` be `d_1`,

`d_1=sqrt((-1-0)^2+(-1-2)^2)`
`color(white)(d_1)=sqrt10`

Let the distance between `(-1,-1)` and `(x,-4)` be `d_2`,

`d_2=sqrt((-1-x)^2+(-1+4)^2)`
`color(white)(d_2)=sqrt((-1-x)^2+9)`

Since, the distance are the same, `d_1=d_2`

`sqrt10=sqrt((-1-x)^2+9)`

Square both sides,

`10=(-1-x)^2+9`

Subtract `9` from both sides,

`(-1-x)^2=1`

Square root both sides,

`-1-x=+-1`

Solve,

`x=0 or -2`

Answer 2

`color(crimson)(x = 0` or `color(crimson)(-2`

Explanation:

Given `A(-1,-1), B(0,2), C(x,-4), bar(AB) = bar(AC)` To find x

Distance formula `d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)`

`bar(AB) = sqrt((-1-0)^2 + (-1-2)^2) = sqrt10`

`bar(AC) = sqrt((-1-x)^2 + (_1+4)^2)`

Bus `bar(AB) = bar(AC)`

`sqrt((-1-x)^2 + 3^2) = sqrt 10`

Squaring both sides,

`(x+1)^2 + 9 = 10`

`(x+1)^2 = 1 = 1^2`

`x+1 = +-1`

`color(crimson)(x = 0, -2`