Point b(-2, 4) lies on a circle centered at a(1, 3). write a paragraph proof to determine whether c(4, 2) also lies on the circle.

we know that

if point B (-2,4) lies on a circle centered at A (1,3)

then

distance AB is equal to the radius of the circle

so

step 1

find the distance AB

dAB=√[(y2-y1)²+(x2-x1)²]

dAB=√[(4-3)²+(1+2)²]> dAB=√[1²+3²]> √10 units

step 2

Find the equation of a circle

(x-h)²+(y-k)²=r²> (x-1)²+(y-3)²=10

step 3

Any point on the circle has to satisfy the equation

(x-1)²+(y-3)² = 10

We replace x and y with the given point C (4,2).

If we replace x with 4 and y with 2

we get

(4-1)² + (2-3)² = 9+1 

10=10

Because both sides of the equation are equal this means that the point

C (4,2) is on the circle.

Any point that is not on the circle won't satisfy the equation.

Only points on the circle will give us the result of 10 = 10

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.