We will learn how to solve different types of problems on ellipse.
1. Find the equation of the ellipse whose eccentricity is \(\frac{4}{5}\) and axes are along the coordinate axes and with foci at (0, ± 4).
Solution:
Let the equitation of the ellipse is \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 ……………… (i)
According to the problem, the coordinates of the foci are (0, ± 4). Therefore, we see that the major axes of the ellipse is along y axes and the minor axes of the ellipse is along x axes.
We know that the coordinates of the foci are (0, ±be).
Therefore, be = 4
b(\(\frac{4}{5}\)) = 4, [Putting the value of e = \(\frac{4}{5}\)]
⇒ b = 5
⇒ b\(^{2}\) = 25
Now, a\(^{2}\) = b\(^{2}\)(1  e\(^{2}\))
⇒ a\(^{2}\) = 5\(^{2}\)(1  (\(\frac{4}{5}\))\(^{2}\))
⇒ a\(^{2}\) = 25(1  \(\frac{16}{25}\))
⇒ a\(^{2}\) = 9
Now putting the value of a\(^{2}\) and b\(^{2}\) in (i) we get, \(\frac{x^{2}}{9}\) + \(\frac{y^{2}}{25}\) = 1.
Therefore, the required equation of the ellipse is \(\frac{x^{2}}{9}\) + \(\frac{y^{2}}{25}\) = 1.
2. Determine the equation of the ellipse whose directrices along y = ± 9 and foci at (0, ± 4). Also find the length of its latus rectum.
Solution:
Let the equation of the ellipse be \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1, ……………………………… (i)
The coordinate of the foci are (0, ± 4). This means that the major axes of the ellipse is along y axes and the minor axes of the ellipse is along x axes.
We know that the coordinates of the foci are (0, ± be) and the equations of directrices are y = ± \(\frac{b}{e}\)
Therefore, \(\frac{b}{e}\) = 9 …………….. (ii)
and be = 4 …………….. (iii)
Now, from (ii) and (iii) we get,
b\(^{2}\) = 36
⇒ b = 6
Now, a\(^{2}\) = b\(^{2}\)(1 – e\(^{2}\))
⇒ a\(^{2}\) = b\(^{2}\)  b\(^{2}\)e\(^{2}\)
⇒ a\(^{2}\) = b\(^{2}\)  (be)\(^{2}\)
⇒ a\(^{2}\) = 6\(^{2}\)  4\(^{2}\), [Putting the value of be = 4]
⇒ a\(^{2}\) = 36  16
⇒ a\(^{2}\) = 20
Therefore, the required equation of the ellipse is \(\frac{x^{2}}{20}\) + \(\frac{y^{2}}{36}\) = 1.
The required length of latus rectum = 2 ∙ \(\frac{a^{2}}{b}\) = 2 ∙ \(\frac{20}{6}\) = \(\frac{20}{3}\) units.
3. Find the equation of the ellipse whose equation of its directrix is 3x + 4y  5 = 0, coordinates of the focus are (1, 2) and the eccentricity is ½.
Solution:
Let P (x, y) be any point on the required ellipse and PM be the perpendicular from P upon the directrix 3x + 4y  5 = 0
Then by the definition,
\(\frac{SP}{PM}\) = e
⇒ SP = e ∙ PM
⇒ \(\sqrt{(x  1)^{2} + (y  2)^{2}}\) = ½ \(\frac{3x + 4y  5}{\sqrt{3^{2}} + 4^{2}}\)
⇒ (x  1)\(^{2}\) + (y  2)\(^{2}\) = ¼ ∙ \(\frac{(3x + 4y  5)^{2}}{25}\), [Squaring both sides]
⇒ 100(x\(^{2}\) + y\(^{2}\) – 2x – 4y + 5) = 9x\(^{2}\) + 16y\(^{2}\) + 24xy  30x  40y + 25
⇒ 91x\(^{2}\) + 84y\(^{2}\)  24xy  170x  360x + 475 = 0, which is the required equation of the ellipse.
`11 and 12 Grade Math
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