Solved on Nov 15, 2023

Rewrite $x^2 + y^2 - 18x - 2y + 1 = 0$ in standard form of a circle equation by completing the square.

STEP 1

Assumptions1. The given equation is $x^{}+y^{}-18x-y+1=0$ . The standard form of a circle equation is $(x-h)^{}+(y-k)^{}=r^{}$ , where (h, k) is the center of the circle and r is the radius.

STEP 2

First, we need to group the x terms and y terms together.
$x^{2}-18x+y^{2}-2y+1=0$

STEP 3

Rearrange the equation to make it easier to complete the square.
$x^{2}-18x+y^{2}-2y=-1$

STEP 4

To complete the square, we need to add and subtract the square of half the coefficient of x and y inside the brackets. For x, half of the coefficient is $-18/2=-9$ , and its square is $81$ . For y, half of the coefficient is $-2/2=-1$ , and its square is $1$ .
$x^{2}-18x+81-81+y^{2}-2y+1-1=-1$

STEP 5

Now, we can rewrite the equation in the form of squares.
$(x-9)^{2}-81+(y-1)^{2}-1=-1$

STEP 6

implify the equation by moving the constants to the right side of the equation.
$(x-9)^{2}+(y-1)^{2}=81+1+1$

STEP 7

Calculate the right side of the equation.
$(x-9)^{2}+(y-1)^{2}=83$ So, the equation of the circle in standard form is $(x-9)^{2}+(y-1)^{2}=83$ . Here, the center of the circle is (9,1) and the radius is $\sqrt{83}$ .

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Rewrite $x^2 + y^2 - 18x - 2y + 1 = 0$ in standard form of a circle equation by completing the square.

STEP 1

Assumptions1. The given equation is $x^{}+y^{}-18x-y+1=0$ . The standard form of a circle equation is $(x-h)^{}+(y-k)^{}=r^{}$ , where (h, k) is the center of the circle and r is the radius.

STEP 2

First, we need to group the x terms and y terms together.
$x^{2}-18x+y^{2}-2y+1=0$

STEP 3

Rearrange the equation to make it easier to complete the square.
$x^{2}-18x+y^{2}-2y=-1$

STEP 4

To complete the square, we need to add and subtract the square of half the coefficient of x and y inside the brackets. For x, half of the coefficient is $-18/2=-9$ , and its square is $81$ . For y, half of the coefficient is $-2/2=-1$ , and its square is $1$ .
$x^{2}-18x+81-81+y^{2}-2y+1-1=-1$

STEP 5

Now, we can rewrite the equation in the form of squares.
$(x-9)^{2}-81+(y-1)^{2}-1=-1$

STEP 6

implify the equation by moving the constants to the right side of the equation.
$(x-9)^{2}+(y-1)^{2}=81+1+1$

STEP 7

Calculate the right side of the equation.
$(x-9)^{2}+(y-1)^{2}=83$ So, the equation of the circle in standard form is $(x-9)^{2}+(y-1)^{2}=83$ . Here, the center of the circle is (9,1) and the radius is $\sqrt{83}$ .

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Rewrite x2+y2−18x−2y+1=0x^2 + y^2 - 18x - 2y + 1 = 0x2+y2−18x−2y+1=0 in standard form of a circle equation by completing the square.

STEP 1

Assumptions1. The given equation is x+y−18x−y+1=0x^{}+y^{}-18x-y+1=0x+y−18x−y+1=0 . The standard form of a circle equation is (x−h)+(y−k)=r(x-h)^{}+(y-k)^{}=r^{}(x−h)+(y−k)=r, where (h, k) is the center of the circle and r is the radius.

STEP 2

First, we need to group the x terms and y terms together.x2−18x+y2−2y+1=0x^{2}-18x+y^{2}-2y+1=0x2−18x+y2−2y+1=0

STEP 3

Rearrange the equation to make it easier to complete the square.x2−18x+y2−2y=−1x^{2}-18x+y^{2}-2y=-1x2−18x+y2−2y=−1

STEP 4

STEP 5

Now, we can rewrite the equation in the form of squares.(x−9)2−81+(y−1)2−1=−1(x-9)^{2}-81+(y-1)^{2}-1=-1(x−9)2−81+(y−1)2−1=−1

STEP 6

implify the equation by moving the constants to the right side of the equation.(x−9)2+(y−1)2=81+1+1(x-9)^{2}+(y-1)^{2}=81+1+1(x−9)2+(y−1)2=81+1+1

STEP 7

Calculate the right side of the equation.(x−9)2+(y−1)2=83(x-9)^{2}+(y-1)^{2}=83(x−9)2+(y−1)2=83So, the equation of the circle in standard form is (x−9)2+(y−1)2=83(x-9)^{2}+(y-1)^{2}=83(x−9)2+(y−1)2=83. Here, the center of the circle is (9,1) and the radius is 83\sqrt{83}83​.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Rewrite $x^2 + y^2 - 18x - 2y + 1 = 0$ in standard form of a circle equation by completing the square.

Assumptions1. The given equation is $x^{}+y^{}-18x-y+1=0$ . The standard form of a circle equation is $(x-h)^{}+(y-k)^{}=r^{}$ , where (h, k) is the center of the circle and r is the radius.

First, we need to group the x terms and y terms together.
$x^{2}-18x+y^{2}-2y+1=0$

Rearrange the equation to make it easier to complete the square.
$x^{2}-18x+y^{2}-2y=-1$

Now, we can rewrite the equation in the form of squares.
$(x-9)^{2}-81+(y-1)^{2}-1=-1$

implify the equation by moving the constants to the right side of the equation.
$(x-9)^{2}+(y-1)^{2}=81+1+1$

Calculate the right side of the equation.
$(x-9)^{2}+(y-1)^{2}=83$ So, the equation of the circle in standard form is $(x-9)^{2}+(y-1)^{2}=83$ . Here, the center of the circle is (9,1) and the radius is $\sqrt{83}$ .