Step-by-step examples of finding the center and radius of circlesExample Show
Find the center and radius of the circle. ???x^2+y^2+24x+10y+160=0??? In order to find the center and radius, we need to change the equation of the circle into standard form, ???(x-h)^2+(y-k)^2=r^2???, where ???h??? and ???k??? are the coordinates of the center and ???r??? is the radius. In order to get the equation into standard form, we have to complete the square with respect to both variables. Grouping ???x???’s and ???y???’s together and moving the constant to the right side, we get ???(x^2+24x)+(y^2+10y)=-160??? Completing the square requires us to take the coefficient on the first degree terms, divide them by ???2???, and then square the result before adding the result back to both sides. The coefficient on the ???x??? term is ???24???, so ???\frac{24}{2}=12??? ???12^2=144??? The coefficient on the ???y??? term is ???10???, so ???\frac{10}{2}=5??? ???5^2=25??? Therefore, we add ???144??? inside the parentheses with the ???x??? terms, ???25??? inside the parenthesis with the ???y??? terms, and we also add ???144??? and ???25??? to the right with the ???-160???. ???(x^2+24x +144)+(y^2+10y+25)=-160 + 144+25??? Factor the parentheses and simplify the right side. ???(x+12)^2+(y+5)^2=9??? Therefore, the center of the circle is at ???(h,k)=(-12,-5)??? and its radius is ???r=\sqrt{9}=3???. Example What is the center and radius of the circle? ???6x^2+6y^2+12x-13=0??? In order to find the center and radius, we need to change the equation of the circle into standard form, ???(x-h)^2+(y-k)^2=r^2???, where ???h??? and ???k??? are the coordinates of the center and ???r??? is the radius. In order to get the equation into standard form, we have to complete the square with respect to ???x???. The ???y??? term is already a perfect square. Let’s begin by collecting like terms and moving the ???-13??? to the right. ???6x^2+12x+6y^2=13??? Our next step is to remove the coefficients of the second degree terms of the ???x??? variable and ???y??? variable. First, we’ll factor out a ???6??? then divide by ???6??? on both sides. ???6(x^2+2x+y^2)=13??? ???x^2+2x+y^2=\frac{13}{6}??? Now complete the square of the ???x??? terms. The ???y??? term is already a perfect square. ???(x^2+2x)+y^2=\frac{13}{6}??? Completing the square requires us to take the coefficient on the first degree term, divide it by ???2???, then square the result before adding the result back to both sides. The coefficient on ???x??? is ???2???, so ???\frac{2}{2}=1??? ???1^2=1??? We’ll therefore add ???1??? to both sides, and get ???(x^2+2x+1)+y^2=\frac{13}{6}+1??? Factor the ???x??? terms and simplify the right hand side. ???(x+1)^2+y^2=\frac{19}{6}??? If you want, you may also write the equation as ???(x+1)^2+(y+0)^2=\frac{19}{6}??? The center of the circle ???(h,k)??? is ???(-1,0)??? and the radius is ???\sqrt{19/6}???. Rule out ???-\sqrt{19/6}??? because a radius can't be negative. Get access to the complete Algebra 2 courseCreated by Krishna Nelaturu Reviewed by Aleksandra Zając, MD Last updated: Oct 18, 2022 This standard equation of a circle calculator is helpful to calculate the standard form of a circle equation using its center coordinates and radius, or any other form of the circle equation. With this simple tool in your hand, you can easily find the circle equation in any form you want! In this article, we shall discuss:
How to write a circle equation in standard form?The standard equation of a circle is given by: (x−a)2+(y−b)2=r2\left(x-a\right)^2 + (y-b)^2 = r^2 where:
We can use this equation to find the standard form from its center and radius or vice versa. How to convert the standard equation into parametric formThe parametric form of a circle equation is given by: x=a+rcos(α) y=b+rsin(α)\begin{align*} x &= a + r\cos(\alpha)\\ y &= b + r\sin(\alpha)\\ \end{align*} where:
It is a straightforward conversion between these two forms; the additional parameter α\alpha is not essential for this conversion, so long as the other variables are known. How to convert the standard equation into general formThe general form of the circle equation is an expansion of its standard form. It can be expressed as: x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx+ Ey+ F = 0 where:
Note that the right-hand side (RHS) of this equation has to be zero. Bring every term to the left-hand side (LHS) and simplify. As this is the expansion of the standard form, we can complete the squares of this expanded form to arrive at the standard equation and establish the following relationships between the various parameters in these two forms: D=−2a E=−2bF=a2+b2−r 2\begin{align*} D &= -2a\\ E &= -2b\\ F &= a^2+b^2-r^2\\ \end{align*} We can use these equations to convert between the standard form and the general form of a circle equation. How to use this standard equation of a circle calculatorYou can use this calculator for more than one thing:
In addition, this calculator will also determine other properties of the circle, like its area and circumference. FAQWhat is the equation of a circle with a center (0,0) and radius of 7?x2+y2 = 49. To find this equation, follow these steps:
How do you determine if a point lies on a circle?To determine whether a point P(px,py) lies on a circle (x-a)2 + (y-b)2 = r2, follow these steps:
Center coordinates and radius Standard form: (x - A)² + (y - B)² = C Parametric form: x = A + r cos(α), y = B + r sin(α) General form: x² + y² + Dx + Ey + F = 0 Arc lengthArea of a circleCircle calc: find c, d, a, r… 5 more How do you write the standard equation of a circle with the center and radius?The equation of a circle written in the form (x−h)2+(y−k)2=r2 where (h,k) is the center and r is the radius. The circle centered at the origin with radius 1; its equation is x2+y2=1.
How do you write the equation of a circle given its radius and the center is at H K?The equation of a circle with (h, k) center and r radius is given by:. (x-h)2 + (y-k)2 = r2. (x-1)2+(y-2)2 = 42. (x2−2x+1)+(y2−4y+4) =16.. X2+y2−2x−4y-11 = 0.. |