⧼exchistory⧽
13 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
Describe and sketch the graph of each of the following equations. Label the foci and endpoints of the major and minor axes.
- [math]\frac{x^2}4 + \frac{y^2}9 = 1[/math]
- [math]\frac{x^2}9 + \frac{y^2}4 = 1[/math]
- [math]\frac{x^2}{169} + \frac{y^2}{144} = 1[/math]
- [math]\frac{x^2}{100} + \frac{y^2}{64} = 1[/math]
- [math]\frac{x^2}{17} + \frac{y^2}{16} = 1[/math].
BBy Bot
Nov 03'24
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[/math]
Write an equation for the ellipse satisfying the given conditions.
- Foci at [math](-5,0)[/math] and [math](5,0)[/math]. Minor axis of length [math]24[/math].
- Center at the origin. Major axis horizontal and of length [math]14[/math], minor axis vertical and of length [math]8[/math].
- Center at the origin. Minor axis vertical and of length [math]4[/math]. Passing though [math](3,1)[/math].
- Foci at [math](-4,0)[/math] and [math](4,0)[/math]. Endpoints of major axis at [math](-5,0)[/math] and [math](5,0)[/math].
- The locus of points the sum of whose distances from [math](0,2)[/math] and [math](0,-2)[/math] is [math]7[/math].
BBy Bot
Nov 03'24
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[/math]
It has been shown that the distance between a point on the ellipse
[[math]]
\frac{x^2}{a^2} + \frac{y^2}{a^2-c^2} = 1
[[/math]]
and the focus [math](c,0)[/math] is [math]\left| \frac{xc}a - a \right|[/math].
- Show that this distance is [math]a - \frac{xc}a[/math] for [math]|x| \leq a[/math].
- Show that the distance between a point on the ellipse and the focus [math](-c,0)[/math] is [math]\left| \frac{xc}a + a \right|[/math] and that the distance is [math]a + \frac{xc}a[/math].
- Show that the sum of the distances from a point on the ellipse to the foci is [math]2a[/math] and hence that the graph of [math]\frac{x^2}{a^2} + \frac{y^2}{a^2-c^2} = 1[/math] contains only those points which satisfy the locus definition.
BBy Bot
Nov 03'24
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[/math]
Describe and sketch the graph of each of the following equations.
- [math]\frac{(x-2)^2}{25} + \frac{(y-4)^2}{9} = 1[/math]
- [math]\frac{(x+3)^2}{16} + \frac{(y-2)^2}{25} = 1[/math]
- [math]\frac{(x+5)^2}{169} + \frac{(y+2)^2}{144} = 4[/math]
- [math]25x^2 + 9(y+3)^2 = 225[/math]
- [math]9x^2 + 4y^2 + 36x - 24y + 36 = 0[/math].
BBy Bot
Nov 03'24
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[/math]
The line segment which passes though a focus, is perpendicular to the major axis, and has its endpoints on the ellipse is called a latus rectum.
- Find the length of a latus rectum of the ellipse [math]4x^2 + 9y^2 = 36[/math].
- Find the length of a latus rectum of the ellipse [math]b^2x^2 + a^2y^2 = a^2b^2[/math]. (Assume that [math]b \lt a[/math].)
- Show that both latera recta of an ellipse are the same length.
BBy Bot
Nov 03'24
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[/math]
Write equations of the directrices and find the eccentricity of each of the following ellipses.
- [math]4x^2 + 9y^2 = 36[/math]
- [math]9x^2 + 4y^2 = 144[/math].
BBy Bot
Nov 03'24
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[/math]
Assume that [math]0 \lt c \lt a[/math].
- lab{3.3.7a} Find the distance between [math](x,y)[/math] and [math](-c,0)[/math].
- lab{3.3.7b} Find the distance between [math](x,y)[/math] and the line [math]x = -\frac{a^2}c[/math].
- Find the locus of points [math](x,y)[/math] such that the ratio between the distance in \ref{ex3.3.7a} and the distance in \ref{ex3.3.7b} is a constant [math]\frac ca[/math].
BBy Bot
Nov 03'24
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[/math]
Show that an ellipse becomes more nearly circular as its foci get closer and closer together.
BBy Bot
Nov 03'24
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[/math]
Consider a point [math](x_1,y_1)[/math] on the graph of [math]b^2x^2 + a^2y^2 = a^2b^2[/math].
- lab{3.3.9a} Find the slope of the tangent to the graph at [math](x_1,y_1)[/math].\
- Write an equation of the tangent line in \ref{ex3.3.9a}.
- Show that [math]b^2xx_1 + a^2yy_1 = a^2b^2[/math] is an equation of the tangent line.
BBy Bot
Nov 03'24
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[/math]
Assume that the constants [math]a[/math], [math]b[/math], [math]c[/math], [math]d[/math], and [math]e[/math] are such that [math]ax^2 + by^2 + cx + dy + e = 0[/math] is an equation of an ellipse. Consider a point [math](x_1,y_1)[/math] on this ellipse.
- lab{3.3.10a} Find the slope of the tangent to the graph at [math](x_1,y_1)[/math].
- Write an equation of the tangent line in \ref{ex3.3.10a}.
- Show that [math]axx_1 + byy_1 + \frac12c(x+x_1) + \frac12d(y+y_1) + e = 0[/math] is an equation of the tangent line.