Conics Formulas

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Last updated 1:39 AM on 6/9/26
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36 Terms

1
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Vertical Parabola Formula

(x−h)2=4p(y−k)\left(x-h\right)^2=4p\left(y-k\right)

2
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Vertical Parabola focus

(h,k+p)\left(h,k+p\right)

3
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Vertical Parabola directrix

y=k−py=k-p

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Horizontal Parabola Formula

(y−k)2=4p(x−h)\left(y-k\right)^2=4p\left(x-h\right)

5
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Horizontal Parabola focus

(h+p,k)\left(h+p,k\right)

6
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Horizontal Parabola directrix

x=h−px=h-p

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Horizontal Ellipse Formula

(x−h)2a2+(y−k)2b2=1\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1

8
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Horizontal Ellipse vertices

(h±a,k)\left(h\pm a,k\right)

9
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Horizontal Ellipse co-vertices

(h,k±b)\left(h,k\pm b\right)

10
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Horizontal ellipse focus

(h±c,k)\left(h\pm c,k\right)

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Vertical Ellipse formula

(y−k)2a2+(x−h)2b2=1\frac{\left(y-k\right)^2}{a^2}+\frac{\left(x-h\right)^2}{b^2}=1

12
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Vertical Ellipse vertices

(h,k±a)\left(h,k\pm a\right)

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Vertical Ellipse co-vertices

(h±b,k)\left(h\pm b,k\right)

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Vertical Ellipse focus

(h,k±c)\left(h,k\pm c\right)

15
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Ellipse Relationship between abc

c2=a2−b2c^2=a^2-b^2

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Eccentricity of Ellipse

e=cae=\frac{c}{a}

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Circle formula

(x−h)2+(y−k)2=r2\left(x-h\right)^2+\left(y-k\right)^2=r^2

18
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Horizontal Hyperbola Formula

(x−h)2a2−(y−k)2b2=1\frac{\left(x-h\right)^2}{a^2}-\frac{\left(y-k\right)^2}{b^2}=1

19
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Horizontal hyperbola vertices

(h±a,k)\left(h\pm a,k\right)

20
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Horizontal hyperbola foci

(h±c,k)\left(h\pm c,k\right)

21
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Horizontal hyperbola asymptotes

(y−k)=±ba(x−h)\left(y-k\right)=\pm\frac{b}{a}\left(x-h\right)

22
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Vertical hyperbola formula

(y−k)2a2−(x−h)2b2=1\frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1

23
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Vertical hyperbola vertices

(h,k±a)\left(h,k\pm a\right)

24
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Vertical hyperbola foci

(h,k±c)\left(h,k\pm c\right)

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Vertical hyperbola asymptotes

(y−k)=±ab(x−h)\left(y-k\right)=\pm\frac{a}{b}\left(x-h\right)

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Hyperbola connection between abc

c2=a2+b2c^2=a^2+b^2

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Discriminant for ellipse

b^2-4ac<0

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Discriminant for parabola

b2−4ac=0b^2-4ac=0

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Discriminant for hyperbola

b^2-4ac>0

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Directrix is perpendicular to polar axis and to the left

r=ed1−ecos⁡θr=\frac{ed}{1-e\cos\theta}

31
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directrix is perpendicular to polar axis and to the right

r=ed1+ecos⁡θr=\frac{ed}{1+e\cos\theta}

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directrix parallel to polar axis and below

r=ed1−esin⁡θr=\frac{ed}{1-e\sin\theta}

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directrix parallel to polar axis and above

r=ed1+esin⁡θr=\frac{ed}{1+e\sin\theta}

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e for parabola

e=1e=1

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e for ellipse

e<1

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e for hyperbola

e>1