Glossary

13.2 Equations of lines and circles in the coordinate plane

3 min read•july 22, 2024

Lines and circles are the building blocks of geometry. They're everywhere, from the roads we drive on to the orbits of planets. Understanding their equations helps us describe and predict their behavior in the world around us.

Equations of lines come in different forms, each useful for specific situations. For circles, we use standard and general forms to describe their position and size. Knowing how to work with these equations opens up a world of geometric possibilities.

Equations of Lines

Equations of lines from given information

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  • y=mx+by = mx + b represents a line with slope mm and y-intercept bb
  • y−y1=m(x−x1)y - y_1 = m(x - x_1) represents a line with slope mm passing through point (x1,y1)(x_1, y_1)
  • y−y1=y2−y1x2−x1(x−x1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) represents a line passing through points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2)
  • have the same slope (m1=m2)(m_1 = m_2)
  • have slopes that are negative reciprocals of each other (m1⋅m2=−1)(m_1 \cdot m_2 = -1)
    • Example: Lines with slopes 22 and −12-\frac{1}{2} are perpendicular

Equations of Circles

Center and radius of circles

  • (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 represents a circle with (h,k)(h, k) and rr
    • Example: (x−3)2+(y+2)2=16(x - 3)^2 + (y + 2)^2 = 16 represents a circle with center (3,−2)(3, -2) and radius 44

Circle equations: standard vs general form

  • x2+y2+Ax+By+C=0x^2 + y^2 + Ax + By + C = 0
  • Converting from general form to standard form:
    1. Complete the square for both xx and yy terms
    2. Rewrite the equation in standard form
  • Converting from standard form to general form:
    1. Expand the squared binomials
    2. Combine like terms to get the equation in general form
    • Example: (x−1)2+(y+3)2=9(x - 1)^2 + (y + 3)^2 = 9 in standard form is equivalent to x2+y2−2x+6y+1=0x^2 + y^2 - 2x + 6y + 1 = 0 in general form

Tangent lines to circles

  • A intersects a circle at exactly one point, called the
  • The radius drawn to the point of tangency is perpendicular to the tangent line
  • Steps to find the :
    1. Find the slope of the radius using the center and the point of tangency
    2. The slope of the tangent line is the of the slope of the radius
    3. Use the point of tangency and the slope of the tangent line to write the equation of the tangent line in point-slope form or slope-intercept form
    • Example: For a circle with center (2,3)(2, 3) and a point of tangency at (5,6)(5, 6), the slope of the radius is 6−35−2=1\frac{6-3}{5-2} = 1, so the slope of the tangent line is −1-1. Using the point-slope form with the point of tangency, the equation of the tangent line is y−6=−1(x−5)y - 6 = -1(x - 5) or y=−x+11y = -x + 11

Key Terms to Review (14)

Center: In geometry, the center refers to a specific point that is equidistant from all points on a given shape, such as a circle. This central point plays a crucial role in defining the properties and equations of circles and is essential for understanding their relationships with other geometric figures. The concept of the center helps in determining various characteristics, such as the radius and diameter, and is fundamental in exploring how circles interact with lines and other circles.
Equation of a tangent line: The equation of a tangent line is a linear equation that touches a curve at exactly one point, providing the slope of the curve at that point. This concept connects closely with the idea of derivatives in calculus, as the slope of the tangent line is equal to the derivative of the function at that point. Understanding this relationship is crucial for analyzing the properties of curves and their intersections with lines, particularly in relation to circles and other geometric shapes.
General form of a circle: The general form of a circle is an algebraic equation that represents all the points in a plane that are equidistant from a fixed center point. This equation is typically expressed as $x^2 + y^2 + Dx + Ey + F = 0$, where D, E, and F are constants that relate to the circle's center and radius. Understanding this form is essential for analyzing and manipulating circles within the coordinate plane, connecting it to the broader context of geometry and algebra.
Midpoint formula: The midpoint formula is a mathematical equation used to find the exact center point between two points in a coordinate plane. It plays a crucial role in geometry, especially in connecting concepts such as distance, line segments, and the equations of shapes like circles. By averaging the x-coordinates and the y-coordinates of two endpoints, the midpoint formula provides essential insights into geometric properties and relationships.
Negative reciprocal: A negative reciprocal refers to two numbers or slopes that, when multiplied together, equal -1. This concept is crucial in understanding relationships between lines, particularly when dealing with parallel and perpendicular lines. If one line has a slope of 'm', the slope of a line that is perpendicular to it will be '-1/m', demonstrating this negative reciprocal relationship.
Parallel Lines: Parallel lines are straight lines in a plane that never meet, no matter how far they are extended, and they maintain a constant distance apart. This concept is crucial for understanding various geometrical relationships, the properties of angles formed when parallel lines intersect with transversals, and for using coordinate geometry to prove line relationships.
Perpendicular lines: Perpendicular lines are two lines that intersect at a right angle, which is 90 degrees. This relationship creates unique properties in geometry, such as the ability to form right triangles and the significance of slope in coordinate geometry, connecting various concepts in mathematics.
Point of tangency: A point of tangency is a specific point where a tangent line touches a curve, such as a circle, without crossing it. This concept is essential in understanding the relationships between lines and circles, as the tangent line is perpendicular to the radius drawn to the point of tangency. The point of tangency serves as a key connection in analyzing tangents, secants, and the properties of circles.
Point-slope form: Point-slope form is a way to express the equation of a line when you know a point on the line and its slope. It is represented as $$y - y_1 = m(x - x_1)$$, where $$m$$ is the slope of the line and $$(x_1, y_1)$$ are the coordinates of the known point. This form is useful for quickly writing equations of lines based on specific points and slopes, which connects to various concepts in geometry and algebra.
Radius: The radius is the distance from the center of a circle or sphere to any point on its boundary. This key measurement is essential in understanding the properties and formulas related to circles, spheres, and other three-dimensional figures, as it directly influences calculations for circumference, area, volume, and surface area.
Slope-intercept form: Slope-intercept form is a way to express the equation of a line using the formula $$y = mx + b$$, where $$m$$ represents the slope of the line and $$b$$ represents the y-intercept, or where the line crosses the y-axis. This format makes it easy to identify key characteristics of a line, such as its steepness and its starting point on the y-axis. By using this form, you can quickly graph a line and understand how it behaves in relation to other lines and shapes on a coordinate plane.
Standard form of a circle: The standard form of a circle is an equation that represents a circle in the coordinate plane, expressed as $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. This form makes it easy to identify key features of the circle, including its center and radius, which are essential for graphing and analyzing circular shapes in geometry.
Tangent Line: A tangent line is a straight line that touches a curve at exactly one point without crossing it. In the context of circles, the tangent line is perpendicular to the radius at the point of contact, showcasing a unique relationship between angles, segments, and the properties of circles. This concept is fundamental in understanding various geometric relationships and equations involving circles and lines.
Two-point form: The two-point form is a method for writing the equation of a line using two distinct points that lie on that line. By knowing the coordinates of these two points, one can derive the slope and subsequently create the equation in slope-intercept or point-slope form. This approach highlights the relationship between linear equations and their graphical representations in the coordinate plane.
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