Circle Equation
Circle Equation - Related Videos
01:18
Circles
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2025
A circle in the coordinate plane is defined as the set of all points that lie at a constant distance, known as the radius, from a fixed point called the center. This relationship is captured using the distance formula. For a point (x, y) on the circle and a center (h, k), the distance between them equals the radius r. By squaring both sides of the distance formula, the equation of the circle is written in standard form:Constructing the Equation from Geometric InformationIf the center and the...
06:30
Protocol for Isolating the Mouse Circle of Willis
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Cited by 4 •
2016
We describe here a reproducible protocol for isolating the mouse circle of Willis.
03:10
Chemical Equations
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2020
Chemical equations represent the identities and relative quantities of substances involved in a chemical reaction. The substances undergoing reaction are called reactants, and their formulas are placed on the left side of the equation. The substances generated by the reaction are called products, and their formulas are placed on the right side of the equation. Plus signs (+) separate individual reactant and product formulas, and an arrow (→) separates the reactant and product (left and right)...
02:55
Thermochemical Equations
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2020
For a chemical reaction (the system) carried out at constant pressure – with the only work done caused by expansion or contraction – the enthalpy of reaction (also called the heat of reaction, ΔHrxn) is equal to the heat exchanged with the surroundings (qp). The change in enthalpy is an extensive property, and it depends on the amounts of the reactants participating in the reaction (or the number of moles of reactants). The change in enthalpy is specific to the reaction, and the physical...
02:59
The Nernst Equation
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2020
Nonstandard Reaction Conditions The interconnection between standard cell potentials and various thermodynamic parameters such as the standard free energy change ΔG° and equilibrium constant K has been previously explored. For example, a redox reaction involving zinc(II) and tin(II) ions at 1 M concentration with Eºcell = +0.291 V and ΔG° = −56.2 kJ is spontaneous. Discharge of this cell, however, results in a change in the reactant concentration and a steady decrease of the cell potential. At...
FAQs
In (x - h)^2 + (y - k)^2 = r^2, the center is (h, k), so the signs inside the parentheses must be interpreted carefully. For example, (x - 3)^2 + (y + 2)^2 = 25 has center (3, -2) and radius 5. Identifying these values immediately supplies the information needed to position and scale the graph.
The squared terms encode the relationship between a point and the center through its horizontal and vertical separations. Squaring makes both directions contribute positively, while their sum equals r^2. This algebraic structure connects the equation to coordinate geometry: instead of measuring distance separately for every point, one relationship describes all locations at the required distance.
Expanded form removes the grouped squares and collects terms, which can make algebraic comparison or solving more convenient. To recover the center and radius, the expression is reorganized by grouping the x-terms and y-terms, then converting each group back into a squared form. This process reveals the geometric parameters that are less obvious in the expanded expression.
The value on the right side determines the squared radius, so a positive value produces a circle with a corresponding positive radius. A larger r^2 places points farther from the center, while a smaller positive value produces a smaller circle. Checking this value helps determine whether the equation describes the intended geometric size before graphing or solving.
First identify h, k, and r from standard form. Plot the center at (h, k), then use the radius to locate points horizontally and vertically from that center. Connecting the resulting circular path gives the graph. If the equation is expanded, convert it first so the center and radius are explicit, reducing the chance of sign or scale errors.
To find where a circle meets another curve or geometric object, both relationships are solved together as simultaneous equations. Substitution or another algebraic method produces coordinate values that satisfy both conditions. The resulting points identify the intersections, allowing the circle to be compared with lines, other equations, or shapes in analytic geometry and applied mathematical models.