Question

Arrange the circles (represented by their equations in general form) in ascending order of their radius lengths.

Tiles
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

Answer 1

There is 2 ways we can know what is the radius of each circles. The first is graphing the equation which I consider tedious. The second is algebraically altering the equation into the form: . Given that, the answer is:x2 + y2 − 2x + 2y − 1 = 0 (r = 1.73)x2 + y2 − 4x + 4y − 10 = 0  (r = 4.24)4x2 + 4y2 + 16x + 24y − 40 = 0 (r = 4.80)x2 + y2 − 8x − 6y − 20 = 0 (r = 6.7)x2 + y2 + 12x − 2y − 9 = 0 (r = 6.78)2x2 + 2y2 − 28x − 32y − 8 = 0 (r = 11)

Answer 2

Answer with explanation:

We know that the standard form of a circle is given by the equation:

`(x-h)^2+(y-k)^2=r^2`

where the center if the circle is at (h,k) and the radius of circle is: r

1)

`x^2+y^2-2x+2y-1=0`

on converting to standard form of equation we get:

`(x-1)^2+(y+1)^2=(√(3))^2`

Hence, the radius of circle is:√3=1.732 units

2)

`x^2+y^2-4x+4y-10=0`

on converting to standard form of equation we get:

`(x-2)^2+(y+2)^2=(√(18))^2`

Hence, the radius of circle is:3√2=4.242 units

3)

`x^2+y^2-8x-6y-20=0`

on converting to standard form of equation we get:

`(x-4)^2+(y-3)^2=(√(45))^2`

Hence, the radius of circle is:3√5=6.708 units

4)

`4x^2+4y^2+16x+24y-40=0`

on dividing both side by 4 we get:

`x^2+y^2+4x+6y-10=0`

on converting to standard form of equation we get:

`(x+2)^2+(y+3)^2=(√(23))^2`

Hence, the radius of circle is:
`√(23)`=4.796 units

5)

`5x^2+5y^2-20x+30y+40=0`

on dividing both side by 5 we get:

`x^2+y^2-4x+6y+8=0`

on converting to standard form of equation we get:

`(x-2)^2+(y+3)^2=(√(5))^2`

Hence, the radius of circle is:√5=2.236 units

6)

`2x^2+2y^2-28x-32y-8 = 0`

on dividing both side by 2 we get:

`x^2+y^2-14x+16y-4=0`

on converting to standard form of equation we get:

`(x-7)^2+(y+8)^2=(√(117))^2`

Hence, the radius of circle is:√117=10.817 units

7)

`x^2+y^2+12x-2y-9=0`

on converting to standard form of equation we get:

`(x+6)^2+(y-1)^2=(\sqrt46})^2`

Hence, the radius of circle is:√46=6.782 units

On arranging the circle in ascending order of their radius is:

1) → 5) → 2) → 4) → 3) → 7) → 6)