Topic
Are the conic lines with the following equations ellipses or hyperbolas? Find the lengths of the axes, the coordinates of the focal point, the focal length, the eccentricity of those conic lines.
a) \(\frac{{{x^2}}}{{100}} + \frac{{{y^2}}}{{64}} = 1\)
b) \(\frac{{{x^2}}}{{36}} – \frac{{{y^2}}}{{64}} = 1\)
Solution method – See details
For ellipse(E): \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) \((0 < b < a)\) . Then we have;
+ Large axis length: \(2a\), minor axis length: \(2b\)
+ Focus \({F_1}( – c;0),{F_2}(c;0)\)
+ Focal length: \(2c = 2\sqrt {{a^2} – {b^2}} \)
+ Miscenter of the ellipse: \(e = \frac{c}{a}\)
Equation of hyperbola \(\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1\” ) where \(a > 0,b > 0\). Then we have:
+ Real axis length: \(2a\), imaginary axis length: \(2b\)
+ Focus \({F_1}( – c;0),{F_2}(c;0)\)
+ Focal length: \(2c = 2\sqrt {{a^2} + {b^2}} \)
+ Miscenter \(e = \frac{c}{a}\)
Detailed explanation
a) \(\frac{{{x^2}}}{{100}} + \frac{{{y^2}}}{{64}} = 1\)
This is the ellipse. We have: \(a = 10,b = 8\)
+ Length of major axis: \(2a = 2.10 = 20\), length of minor axis: \(2b = 2.8 = 16\)
+ Focal length: \(2c = 2\sqrt {{a^2} – {b^2}} = 2\sqrt {{{10}^2} – {8^2}} = 2.6 = 12\)
+ Focus \({F_1}( – 6;0),{F_2}(6;0)\)
+ Miscenter of ellipse: \(e = \frac{6}{{10}} = 0.6\)
b) \(\frac{{{x^2}}}{{36}} – \frac{{{y^2}}}{{64}} = 1\)
This is the hyperbolic curve. We have: \(a = 6,b = 8\)
+ Real axis length: \(2a = 2.6 = 12\), imaginary axis length: \(2b = 2.8 = 16\)
+ Focal length: \(2c = 2\sqrt {{a^2} + {b^2}} = 2\sqrt {{6^2} + {8^2}} = 2.10 = 20\)
+ Focus \({F_1}( – 10;0),{F_2}(10;0)\)
+ Wrong center \(e = \frac{{10}}{6} = \frac{5}{3}\)
