91Ó°ÊÓ

To understand how to determine the equation of a circle, we first need to grasp the concept of the "standard form of a circle". The standard form is written as \((x-h)^2 + (y-k)^2 = r^2\), where:

• \( (h, k) \) is the center of the circle.
• \( r \) is the radius of the circle.

This form is particularly useful because it easily shows both the center and the radius, allowing for quick interpretation and graphing of the circle. Transforming any given quadratic equation in terms of \( x \) and \( y \) into this form can reveal whether it represents a true circle or not.

The equation \((x-h)^2 + (y-k)^2 = r^2\) describes a circle in a 2-dimensional plane. Here, \( (h, k) \) designates the fixed point or center from which every point on the circle is equidistant. The distance, denoted by \( r \), is the radius. Such an equation ensures that all the point coordinates \((x, y)\) at the calculated distance \( r \) from \( (h, k) \) make the circle complete.
For circles, an absence of an \( x \, y \) term (like +2xy in equations of ellipses or hyperbolas) keeps the shape uniform and round, maintaining perfect symmetry in all directions from the center. Identifying whether an equation is a true circle requires us to put it in its standard form and evaluate the symmetry of its terms.

Determining the center and radius of a circle from its equation is simple once the equation is in its standard form. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the center \( (h, k) \) can be directly read as the values of \( h \) and \( k \) (keeping in mind the signs change). The radius \( r \) is the square root of the right side constant, \( r^2 \). For instance, if the transformed equation is \((x + \frac{1}{2})^2 + (y - \frac{3}{5})^2 = \frac{161}{100}\), comparing it to the standard form gives us:

• Center: \( (-\frac{1}{2}, \frac{3}{5}) \)
• Radius: \( r = \sqrt{\frac{161}{100}} = \frac{\sqrt{161}}{10} \)

Understanding these values helps in visualizing the size and location of the circle in a coordinate plane.

Algebraic transformation, such as completing the square, is a technique used to convert quadratic equations into a form where properties of geometric shapes, like circles, become evident. Completing the square involves adjusting quadratic terms to create perfect square trinomials. This makes it easier to transform a general quadratic equation into its standard form.
For example, given an equation \(x^2 + x + y^2 - \frac{6}{5} y = 1\), group and complete the square for \( x \) and \( y \) terms separately:

• For \( x^2 + x \), add and subtract \( (\frac{1}{2})^2 \).
• For \( y^2 - \frac{6}{5} y \), add and subtract \( (\frac{3}{5})^2 \).

Through this algebraic manipulation, balance the equation with constants, thus allowing the rearrangement into a clear and interpretable form to identify geometric properties like the circle's center and radius.