How do you find the value of a given the points (4,-1), (a,5) with a distance of 10?

2 Answers
Apr 11, 2018

color(blue)(a = 12, -4

Explanation:

Distance between two points is given by the formula :

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

"Given " (x_1, y_1) = (4,-1), (x_2,y_2) = (a,5), d = 10, " To find " a

10 = sqrt ((a - 4)^2 + (5+1)^2)

(a-4)^2 + 6^2 = 10^2, " squaring both sides"

(a - 4)^2 = 10^2 - 6^2 = 64 = 8^2

a - 4 = +- 8 , " taking root on both sides"

color(brown)(a =) +-8 + 4 = color(brown)(12, -4

Apr 11, 2018

a=12, a=-4

Explanation:

First, let's take a look at the distance formula where d is the distance

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

Now you choose which point is point 2 (includes y_2 and x_2) and which point is point 1 (includes y_1 and x_1)

Point 2: (a,5)->y_2 = 5 and x_2 =a
Point 1: (4,-1)->y_1 = -1 and x_1 = 4

Now plug these values into the equation

d=sqrt((a-4)^2+(5-(-1))^2)

We also know that the distance d is 10 so we can plug that in

10=sqrt((a-4)^2+(5-(-1))^2)

Simplify

10=sqrt((a-4)^2+(5-(-1))^2)

10=sqrt((a-4)^2+(5+1)^2)

10=sqrt((a-4)^2+(6)^2)

10=sqrt((a-4)^2+36)

Now square both sides to get rid of the radical

10^2=(sqrt((a-4)^2+36))^2

100=(a-4)^2+36

FOIL the expression (a-4)^2

(a-4)^2

(a-4)(a-4)

a^2+a(-4)+a(-4)+(-4)(-4)

a^2-4a-4a+16

a^2-8a+16

Now plug this expression back in for (a-4)^2

100=(a-4)^2+36

100=a^2-8a+16+36

100=a^2-8a+52

Subtract 100 from both sides

a^2-8a+52-100=100-100

a^2-8a-48=0

Now factor this expression to find the roots/zeros

a^2-8a-48=0

(a-12)(a+4)=0

So the roots are

a=12, a=-4