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

2 Answers
Sonnhard
Jun 24, 2018

#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#

Jim G.
Jun 24, 2018

#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#

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