Question

How do you find the possible values for a if the points (-5,a), (3,1) has a distance of `sqrt89`?

Answer 1

`a=6` or `a=-4`

Explanation:

The formula for distance of two Points is given by

`d(P_1,P_2)=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`
so we get

`sqrt((-5-3)^2+(a-1)^2)=sqrt(89)`

squaring we get

`64+a^2-2a+1=89`
combining like Terms

`a^2-2a-24=0`

using the quadratic formula we get

`a_(1,2)=1+pmsqrt(25)`

so
`a_1=6`
`a_2=-4`

Answer 2

`a=-4" or "a=6`

Explanation:

`"to calculate the distance d use the "color(blue)"distance formula"`

`•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`"let "(x_1,y_1)=(3,1)" and "(x_2,y_2)=(-5,a)`

`d=sqrt((-5-3)^2+(a-1)^2)=sqrt89`

`sqrt(64+(a-1)^2)=sqrt89`

`color(blue)"square both sides"`

`64+(a-1)^2=89`

`"subtract 64 from both sides"`

`(a-1)^2=25`

`color(blue)"take the square root of both sides"`

`sqrt((a-1)^2)=+-sqrt25larrcolor(blue)"note plus or minus"`

`a-1=+-5`

`"add 1 to both sides"`

`a=1+-5`

`a=1-5=-4" or "a=1+5=6`