Background.  

    Five points in the plane uniquely determine an equation for a conic section. The implicit formula for a conic section is often mentioned in textbooks, and the special cases for an ellipse, hyperbola, parabola, circle are obtained by either setting some coefficients equal to zero or making them the same value.

Implicit Equation for a Line.  

    The equation    of the line through the two points    can be computed with the determinant    

          

Example 1.  Use the determinant method to find the line through the points (1,4) and {5,3).
Solution 1.

Example 2.  Use the determinant method to find the line through the points  (3,1) and (3,4).
Solution 2.

Example 3.  Use the determinant method to find the line through the points  (1,4) and (5,4).
Solution 3.

Implicit Equation for a Circle.  

    The equation of the circle    through the three points    can be computed with the determinant    

          

Example 4.  Use the determinant method to find the circle through the points (6,1),  (2,2)  and  (1,4).
Remark. In Exercises 5 and 6 the same points are used to find the standard parabola and alternate parabola.
Solution 4.

Implicit Equation for a Parabola.  

    The equation of the parabola    through the three points    can be computed with the determinant    

          

Alternate Equation of a Parabola.  

    The equation of the parabola    through the three points    can be computed with the determinant    

          

Example 5.  Use the determinant method to find the standard equation of a parabola through the points (6,1),  (2,2)  and  (1,4).
Remark. In Exercises 4 and 6 the same points are used to find the circle and alternate parabola.
Solution 5.

Example 6.  Use the determinant method to find the alternate equation of a parabola through the points (6,1),  (2,2)  and  (1,4).
Remark. In Exercises 4 and 5 the same points are used to find the circle and standard parabola.
Solution 6.

Implicit Equation for a Standard Ellipse.  

    The equation of the ellipse    through the four points    can be computed with the determinant    

          

Example 7.  Use the determinant method to find the standard ellipse through the points (6,1), (2,2), (1,4), (9,2).
Solution 7.

The Implicit Equation for a 5 Point Conic.  

    The equation of the conic through the five points    can be computed with the determinant    

          

Example 8.  Determine the conic that passes through the five points  .  
Solution 8.

Example 9.  Determine the conic that passes through the five points .  
Solution 9.

Example 10.  Determine the conic that passes through the five points   .  
Solution 10.

Example 11.  Determine the conic that passes through the five points  .  
Solution 11.

Example 12.  Determine the conic that passes through the five points .  
Solution 12.

    Details for the implicit formulas for the line, circle, parabola, ellipse, and conic are given below.  

Proof  Conic Fit  Conic Fit  

Computer Programs  Conic Fit  Conic Fit  

Old Lab Project (Conic Section Curves  Conic Section Curves).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Conic Fit  Conic Fit  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Determinants and Conic Section Curves

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(c) John H. Mathews 2004