3. the standard equation of a circle (not centered at the origin) with radius r can be derived from its locus definition: the set of all points, p(x, y), that are a given distance, r, from a fixed point c(h, k). how could we use the distance formula and the locus definition to derive the standard equation of a circle?

The  equation for  circle of radius centered at the point is:

Step-by-step explanation:

To derive this equation, we need the formula for the distance between two points in the plane. If the point is given by its cartesian coordinates, we can use the Pythagorean theorem to find what the distance between the points is.

Lets say we have two points, one given by cartesian coordinates P:(x,y) and the other given by cartesian coordiantes C:(h,k)

(see diagram)

We can see that

and by the pythagorean theorem we have:

Where is the distance between the points

Now, combining the last three equations we find:

We know that the definition of a circle is the locus of points that are at a distance r from a point called the center of the circle. if we take C:(h,k) as the center of the circle, and , the last equation becomes:

And that is the result we wanted.

answer:

it is general formula that is used to write value to its excet position

a11 a12 a13

a21 a22 a23

it is the example of order 2 by 3 matrix in this

a11 means value occur in 1 st row 1st column

a12 means value occurs in 1st row 2nd column

a13 means value occur in 1st row and 3rd column

a21 means value occurs in 2nd row and 1st column

a22 means value occurs in 2nd row and in 2nd column

a23 means value occurs in 2nd row and in 3rd column