arrange the circles (represented by their equations in general form) in ascending order of their radius lengths.
x2 + y2 − 2x + 2y − 1 = 0
x2 + y2 − 4x + 4y − 10 = 0
x2 + y2 − 8x − 6y − 20 = 0
4x2 + 4y2 + 16x + 24y − 40 = 0
5x2 + 5y2 − 20x + 30y + 40 = 0
2x2 + 2y2 − 28x − 32y − 8 = 0
x2 + y2 + 12x − 2y − 9 = 0

  1, 5, 2, 4, 3, 7, 6

Step-by-step explanation:

After dividing by the leading coefficient, each equation can be put into the form ...

  x² + y² + ax +by +c = 0

Subtracting c and separately completing the square for x-terms and y-terms, we get ...

  x² + ax + (a/2)² + y² + by + (b/2)² = -c + (a/2)² + (b/2)²

  (x +a/2)² + (y +b/2)² = r² = (a/2)² + (b/2)² -c . . . . . rewrite in standard form

Ordering by the square of the radius length will match the ordering by radius length, so we just need to compute (a/2)² +(b/2)² -c for each given equation. I find it convenient to let a calculator or spreadsheet do this calculation (see attached).

In the order the equations are given, the square of the radius is ...

  3, 18, 45, 23, 5, 117, 46

So the order of the equations from smallest radius to largest is ...

  1, 5, 2, 4, 3, 7, 6

arithmetic sequence

step-by-step explanation:

given that a sequence has terms as

19, 15, 11,

this is represented on a graph

the pattern indicates that the sequence is an arithmetic sequence with i term

a=19 and common difference d =-4

any general term

thus the sequence is reduced by 4 for each successive term starting from 1 and nth term is

24-5n.

first negative term would start after 5th term

5th term =24-25< 0

from 5th term onwards the sequence would be negative.