point problems

Question

1) point (7,-6) and (9,4) are end points of the diameter of a circle.

a) What is the length of the diameter? give the exact answer simplify as much as possible show work.

b) What is the center of the circle?

c) What is the equation of the circle?

 

2) Find the equation for a line that passes through the points (7,8) and (9,-4). Write the equation in slope intercept form and show work.

Answer 1

1-part a)

make a sketch and you can see that if you make a right triangle with these two points the diameter is the hypotenuse of the triangle with sides Δx and Δy

so use Pythagoras's Theorem to find the diameter

Diameter = √ (Δx² + Δy²)

you should be able to calculate from here

 

part b)

the center of the circle will lay on the Diameter 1/2 way between the ends

so the coordinates of the center are (a,b)

where a is 1/2 way between the two x coordinates 7 and 9  => 8

and b is 1/2 way between the two y coordinates -6 and 4 =>  you can calculate this

 

part c)

where (a,b) is the center of a circle and r = radius = 1/2 D

the standard form of the equation for a circle is:

(x-a)² + (y-b)² = r²

so just put the values you know into the equation.

 

2-

the standard form of an equation in point-slope form is

(y-y₁) = m (x-x₁)  where m is the slope and (x₁,y₁) is one of the points

So find the slope from Δy/Δx

= (y₂-y₁) / (x₂-x₁)

once you know the slope put that in the standard form for m

and pick one of the points (x,y) to put in the std. form for x₁ and y₁