Key takeaway: This precalculus conic sections lesson shows that every circle, parabola, ellipse and hyperbola is defined by distances. A circle keeps one distance from its center, a parabola is equally far from a focus and a line, an ellipse fixes a sum of two distances, and a hyperbola fixes a difference. Standard G-GPE.A.2 covers the parabola, and G-GPE.A.3 covers the other two. Each section below pairs a definition with a quick example and then with the matching equation.
Conic sections are curves defined by distances. A circle keeps a constant distance from its center, a parabola is equally far from a focus and a directrix, an ellipse has a constant sum of distances to two foci, and a hyperbola has a constant difference. Completing the square converts a general equation to standard form to identify each one.
Precalculus Conic Sections Lesson: key facts
- Circle: \((x-h)^{2}+(y-k)^{2}=r^{2}\), from the distance formula.
- Parabola: \(x^{2}=4py\) has focus \((0,\ p)\) and directrix \(y=-p\).
- Ellipse: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(c^{2}=a^{2}-b^{2}\); the distance sum is \(2a\).
- Hyperbola: \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with \(c^{2}=a^{2}+b^{2}\) and asymptotes \(y=\pm\frac{b}{a}x\).
- Complete the square in \(x\) and \(y\) to rewrite a general conic equation in standard form.
What is the equation of a circle?
A circle is the set of points at a fixed distance \(r\) from a center \((h,\ k)\). The distance formula, which comes from the Pythagorean theorem, gives \((x-h)^{2}+(y-k)^{2}=r^{2}\). The circle below has center \((2,\ -1)\) and passes through \((6,\ 2)\), so \(r=\sqrt{4^{2}+3^{2}}=5\) and its equation is \((x-2)^{2}+(y+1)^{2}=25\).
How is a parabola defined by a focus and a directrix?
A parabola is the set of points equally far from a fixed point, the focus, and a fixed line, the directrix. With vertex at the origin and focus \((0,\ p)\), the directrix is \(y=-p\) and the equation is \(x^{2}=4py\). For focus \((0,\ 2)\), \(p=2\), so \(x^{2}=8y\). The point \((4,\ 2)\) is \(4\) units from the focus and \(2-(-2)=4\) units from the directrix.
What defines an ellipse and a hyperbola?
Ellipse: the sum of the distances to two foci is constant, \(2a\). Standard form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a>b\) and \(c^{2}=a^{2}-b^{2}\); the foci are \((\pm c,\ 0)\).
Hyperbola: the absolute difference of the distances is constant, \(2a\). Standard form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with \(c^{2}=a^{2}+b^{2}\); the foci are \((\pm c,\ 0)\) and the asymptotes are \(y=\pm\frac{b}{a}x\).
For \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\): \(c^{2}=25-9=16\), so the foci are \((\pm 4,\ 0)\) and the point \((3,\ 2.4)\) has distances \(7.4\) and \(2.6\), which add to \(10=2a\).
For \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\): \(c^{2}=16+9=25\), so the foci are \((\pm 5,\ 0)\). The point \((5,\ 2.25)\) is \(2.25\) from one focus and \(10.25\) from the other, a difference of \(8=2a\).
How do you identify a conic by completing the square?
- Group the \(x\) terms and the \(y\) terms.
- Complete the square in each group, adding the same amounts to both sides.
- Divide so the right side is \(1\), then read the type: equal positive squares give a circle; a sum with different denominators gives an ellipse; a difference gives a hyperbola; only one squared variable gives a parabola.
Example. \(x^{2}+y^{2}-6x+4y-12=0\) becomes \((x-3)^{2}+(y+2)^{2}=12+9+4=25\): a circle with center \((3,\ -2)\) and radius \(5\).
Conic Sections: Parabolas, Ellipses and Hyperbolas: frequently asked questions
What is a conic section?
A conic section is a curve formed by slicing a cone with a plane: a circle, ellipse, parabola or hyperbola. Each can also be defined by a distance rule involving a center, a focus, or two foci.
How do you find the focus of a parabola?
Write the equation as \(x^{2}=4py\) or \(y^{2}=4px\) and read \(p\). The focus is \(p\) units from the vertex inside the curve, and the directrix is \(p\) units on the other side.
How can you tell an ellipse from a hyperbola in an equation?
Both have \(x^{2}\) and \(y^{2}\) terms. With a plus sign (and different positive denominators) it is an ellipse. With a minus sign it is a hyperbola.
Why is c^2 = a^2 – b^2 for an ellipse but a^2 + b^2 for a hyperbola?
For an ellipse the foci lie inside the vertices, so \(c < a\). For a hyperbola the foci lie beyond the vertices, so \(c > a\). The distance definitions give these two relations.
How do you complete the square for a conic?
Group the \(x\) terms and the \(y\) terms, add the needed constants to both sides to form perfect squares, and divide to make the right side \(1\) when the conic is an ellipse or hyperbola.
How does a distance rule give the equation of a circle?
Every point \((x,y)\) on a circle lies \(r\) units from the center \((h,k)\). The distance formula then yields \((x-h)^{2}+(y-k)^{2}=r^{2}\). For a circle centered at \((3,-2)\) with radius 4, the equation is \((x-3)^{2}+(y+2)^{2}=16\). Notice the signs flip inside the parentheses, because the equation subtracts the center.
What do the focus and directrix do in a parabola?
A parabola collects every point whose distance to the focus equals its distance to the directrix. For \(x^{2}=12y\), the number \(4p=12\) gives \(p=3\), so the focus is \((0,3)\) and the directrix is \(y=-3\). A satellite dish uses this property, since signals bounce toward the focus.
What separates an ellipse from a hyperbola?
An ellipse fixes the sum of the two focal distances at \(2a\), and it uses \(c^{2}=a^{2}-b^{2}\). A hyperbola fixes the difference and uses \(c^{2}=a^{2}+b^{2}\). In \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\), the minus sign signals a hyperbola, so the asymptotes are \(y=\pm\frac{3}{4}x\).
How can completing the square reveal the conic?
Collect x terms together and y terms together, then complete each square. For \(4x^{2}+y^{2}-8x+6y+9=0\), the result is \(4(x-1)^{2}+(y+3)^{2}=4\). Both squares are positive with different coefficients, so the curve is an ellipse centered at \((1,-3)\).
What is a good next step?
Draw each conic by hand once, marking the foci and the key points. Then write the distance rule next to the picture in your own words. When you can state the rule and sketch the shape, the equations stop feeling like separate formulas.
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Precalculus Conic Sections: Parabolas, Ellipses and Hyperbolas lesson
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