Derive Equations of Ellipses and Parabolas - Geometry
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Find $e$ for a hyperbola with $a=5$ and $c=13$.
Find $e$ for a hyperbola with $a=5$ and $c=13$.
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$e=\frac{13}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{13}{5}$.
$e=\frac{13}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{13}{5}$.
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Find $c$ for an ellipse if $a=13$ and $b=5$.
Find $c$ for an ellipse if $a=13$ and $b=5$.
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$c=12$. Using $c^2 = a^2 - b^2$: $c^2 = 169 - 25 = 144$, so $c = 12$.
$c=12$. Using $c^2 = a^2 - b^2$: $c^2 = 169 - 25 = 144$, so $c = 12$.
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Find $c$ for a hyperbola if $a=8$ and $b=6$.
Find $c$ for a hyperbola if $a=8$ and $b=6$.
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$c=10$. Using $c^2 = a^2 + b^2$: $c^2 = 64 + 36 = 100$, so $c = 10$.
$c=10$. Using $c^2 = a^2 + b^2$: $c^2 = 64 + 36 = 100$, so $c = 10$.
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Find $b^2$ for an ellipse if $a=10$ and $c=6$.
Find $b^2$ for an ellipse if $a=10$ and $c=6$.
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$b^2=64$. Using $c^2 = a^2 - b^2$: $b^2 = 100 - 36 = 64$.
$b^2=64$. Using $c^2 = a^2 - b^2$: $b^2 = 100 - 36 = 64$.
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What is the locus definition of an ellipse using foci $F_1,F_2$ and constant $2a$?
What is the locus definition of an ellipse using foci $F_1,F_2$ and constant $2a$?
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$d(P,F_1)+d(P,F_2)=2a$. The sum of distances from any point on an ellipse to both foci equals $2a$.
$d(P,F_1)+d(P,F_2)=2a$. The sum of distances from any point on an ellipse to both foci equals $2a$.
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What is the locus definition of a hyperbola using foci $F_1,F_2$ and constant $2a$?
What is the locus definition of a hyperbola using foci $F_1,F_2$ and constant $2a$?
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$|d(P,F_1)-d(P,F_2)|=2a$. The absolute difference of distances from any point on a hyperbola to both foci equals $2a$.
$|d(P,F_1)-d(P,F_2)|=2a$. The absolute difference of distances from any point on a hyperbola to both foci equals $2a$.
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What is the distance formula between $(x,y)$ and $(h,k)$ in the plane?
What is the distance formula between $(x,y)$ and $(h,k)$ in the plane?
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$d=\sqrt{(x-h)^2+(y-k)^2}$. The Euclidean distance formula derived from the Pythagorean theorem.
$d=\sqrt{(x-h)^2+(y-k)^2}$. The Euclidean distance formula derived from the Pythagorean theorem.
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What is the standard form of a horizontal ellipse centered at $(h,k)$ with $a>b$?
What is the standard form of a horizontal ellipse centered at $(h,k)$ with $a>b$?
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$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$. For horizontal ellipses, $a$ (semi-major axis) goes under the $x$-term when $a>b$.
$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$. For horizontal ellipses, $a$ (semi-major axis) goes under the $x$-term when $a>b$.
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What is the standard form of a vertical ellipse centered at $(h,k)$ with $a>b$?
What is the standard form of a vertical ellipse centered at $(h,k)$ with $a>b$?
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$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$. For vertical ellipses, $a$ (semi-major axis) goes under the $y$-term when $a>b$.
$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$. For vertical ellipses, $a$ (semi-major axis) goes under the $y$-term when $a>b$.
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What is the standard form of a vertical hyperbola centered at $(h,k)$?
What is the standard form of a vertical hyperbola centered at $(h,k)$?
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$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$. Vertical hyperbolas have the positive term with $y$ and negative term with $x$.
$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$. Vertical hyperbolas have the positive term with $y$ and negative term with $x$.
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For an ellipse, what relationship connects $a$, $b$, and focal distance $c$?
For an ellipse, what relationship connects $a$, $b$, and focal distance $c$?
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$c^2=a^2-b^2$. For ellipses, $c<a$ so $c^2 = a^2 - b^2$ where $c$ is the focal distance.
$c^2=a^2-b^2$. For ellipses, $c<a$ so $c^2 = a^2 - b^2$ where $c$ is the focal distance.
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For a hyperbola, what relationship connects $a$, $b$, and focal distance $c$?
For a hyperbola, what relationship connects $a$, $b$, and focal distance $c$?
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$c^2=a^2+b^2$. For hyperbolas, $c>a$ so $c^2 = a^2 + b^2$ where $c$ is the focal distance.
$c^2=a^2+b^2$. For hyperbolas, $c>a$ so $c^2 = a^2 + b^2$ where $c$ is the focal distance.
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What are the foci of a horizontal ellipse centered at $(h,k)$?
What are the foci of a horizontal ellipse centered at $(h,k)$?
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$(h\pm c,,k)$. Horizontal ellipses have foci along the $x$-axis, displaced $c$ units from center.
$(h\pm c,,k)$. Horizontal ellipses have foci along the $x$-axis, displaced $c$ units from center.
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What are the foci of a vertical ellipse centered at $(h,k)$?
What are the foci of a vertical ellipse centered at $(h,k)$?
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$(h,,k\pm c)$. Vertical ellipses have foci along the $y$-axis, displaced $c$ units from center.
$(h,,k\pm c)$. Vertical ellipses have foci along the $y$-axis, displaced $c$ units from center.
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What are the foci of a horizontal hyperbola centered at $(h,k)$?
What are the foci of a horizontal hyperbola centered at $(h,k)$?
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$(h\pm c,,k)$. Horizontal hyperbolas have foci along the $x$-axis, displaced $c$ units from center.
$(h\pm c,,k)$. Horizontal hyperbolas have foci along the $x$-axis, displaced $c$ units from center.
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What are the foci of a vertical hyperbola centered at $(h,k)$?
What are the foci of a vertical hyperbola centered at $(h,k)$?
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$(h,,k\pm c)$. Vertical hyperbolas have foci along the $y$-axis, displaced $c$ units from center.
$(h,,k\pm c)$. Vertical hyperbolas have foci along the $y$-axis, displaced $c$ units from center.
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Which equation matches a vertical ellipse with center $(h,k)$, $a=7$, and $b=2$?
Which equation matches a vertical ellipse with center $(h,k)$, $a=7$, and $b=2$?
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$\frac{(x-h)^2}{4}+\frac{(y-k)^2}{49}=1$. Vertical ellipse with $a^2 = 49$ and $b^2 = 4$.
$\frac{(x-h)^2}{4}+\frac{(y-k)^2}{49}=1$. Vertical ellipse with $a^2 = 49$ and $b^2 = 4$.
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Which equation matches a horizontal hyperbola with center $(h,k)$, $a=3$, and $b=4$?
Which equation matches a horizontal hyperbola with center $(h,k)$, $a=3$, and $b=4$?
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$\frac{(x-h)^2}{9}-\frac{(y-k)^2}{16}=1$. Horizontal hyperbola with $a^2 = 9$ and $b^2 = 16$.
$\frac{(x-h)^2}{9}-\frac{(y-k)^2}{16}=1$. Horizontal hyperbola with $a^2 = 9$ and $b^2 = 16$.
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What is the asymptote form for a vertical hyperbola $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$?
What is the asymptote form for a vertical hyperbola $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$?
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$y-k=\pm \frac{a}{b}(x-h)$. For vertical hyperbolas, slopes are $\pm \frac{a}{b}$.
$y-k=\pm \frac{a}{b}(x-h)$. For vertical hyperbolas, slopes are $\pm \frac{a}{b}$.
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What is the asymptote form for a horizontal hyperbola $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$?
What is the asymptote form for a horizontal hyperbola $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$?
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$y-k=\pm \frac{b}{a}(x-h)$. For horizontal hyperbolas, slopes are $\pm \frac{b}{a}$.
$y-k=\pm \frac{b}{a}(x-h)$. For horizontal hyperbolas, slopes are $\pm \frac{b}{a}$.
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Find the vertices of the hyperbola with foci $(0,\pm 13)$ and constant difference $10$.
Find the vertices of the hyperbola with foci $(0,\pm 13)$ and constant difference $10$.
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$(0,,\pm 5)$. From constant difference 10: $a = 5$, so vertices are at $(0, \pm 5)$.
$(0,,\pm 5)$. From constant difference 10: $a = 5$, so vertices are at $(0, \pm 5)$.
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What is the equation of the hyperbola with foci $ (5,-4) $ and $ (5,8) $ and constant difference $ 8 $?
What is the equation of the hyperbola with foci $ (5,-4) $ and $ (5,8) $ and constant difference $ 8 $?
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$\frac{(y-2)^2}{16}-\frac{(x-5)^2}{20}=1$. Center $ (5,2) $, $ c = 6 $, $ a = 4 $, vertical, so $ b^2 = 36 - 16 = 20 $.
$\frac{(y-2)^2}{16}-\frac{(x-5)^2}{20}=1$. Center $ (5,2) $, $ c = 6 $, $ a = 4 $, vertical, so $ b^2 = 36 - 16 = 20 $.
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Find the vertices of the ellipse with foci $(\pm 3,0)$ and constant sum $10$.
Find the vertices of the ellipse with foci $(\pm 3,0)$ and constant sum $10$.
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$(\pm 5,,0)$. From constant sum 10: $a = 5$, so vertices are at $(\pm 5, 0)$.
$(\pm 5,,0)$. From constant sum 10: $a = 5$, so vertices are at $(\pm 5, 0)$.
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What are the vertices of a vertical hyperbola centered at $(h,k)$?
What are the vertices of a vertical hyperbola centered at $(h,k)$?
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$(h,,k\pm a)$. Vertices are the endpoints of the transverse axis, $a$ units from center.
$(h,,k\pm a)$. Vertices are the endpoints of the transverse axis, $a$ units from center.
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What are the vertices of a horizontal hyperbola centered at $(h,k)$?
What are the vertices of a horizontal hyperbola centered at $(h,k)$?
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$(h\pm a,,k)$. Vertices are the endpoints of the transverse axis, $a$ units from center.
$(h\pm a,,k)$. Vertices are the endpoints of the transverse axis, $a$ units from center.
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What are the vertices of a vertical ellipse centered at $(h,k)$?
What are the vertices of a vertical ellipse centered at $(h,k)$?
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$(h,,k\pm a)$. Vertices are the endpoints of the major axis, $a$ units from center.
$(h,,k\pm a)$. Vertices are the endpoints of the major axis, $a$ units from center.
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What are the vertices of a horizontal ellipse centered at $(h,k)$?
What are the vertices of a horizontal ellipse centered at $(h,k)$?
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$(h\pm a,,k)$. Vertices are the endpoints of the major axis, $a$ units from center.
$(h\pm a,,k)$. Vertices are the endpoints of the major axis, $a$ units from center.
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Find $e$ for a hyperbola with $a=5$ and $c=13$.
Find $e$ for a hyperbola with $a=5$ and $c=13$.
Tap to reveal answer
$e=\frac{13}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{13}{5}$.
$e=\frac{13}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{13}{5}$.
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Find $e$ for an ellipse with $a=10$ and $c=6$.
Find $e$ for an ellipse with $a=10$ and $c=6$.
Tap to reveal answer
$e=\frac{3}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{6}{10} = \frac{3}{5}$.
$e=\frac{3}{5}$. Eccentricity: $e = \frac{c}{a} = \frac{6}{10} = \frac{3}{5}$.
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What condition on eccentricity $e$ indicates a hyperbola?
What condition on eccentricity $e$ indicates a hyperbola?
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$e>1$. Hyperbolas have eccentricity greater than 1.
$e>1$. Hyperbolas have eccentricity greater than 1.
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