Here you will learn some ellipse examples for better understanding of ellipse concepts.
Example 1 : Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \(\sqrt{5}\)
Solution : Here S = (2, 3) & S’ is (-2, 3) and b = \(\sqrt{5}\) \(\implies\) SS’ = 4 = 2ae \(\implies\) ae = 2
but \(b^2\) = \(a^2(1-e^2)\) \(\implies\) 5 = \(a^2\) – 4 \(\implies\) a = 3
Hence the equation to major axis is y = 3.
Centre of ellipse is midpoint of SS’ i.e. (0, 3)
\(\therefore\) Equation to ellipse is \(x^2\over a^2\) + \({(y-3)}^2\over b^2\) = 1 or \(x^2\over 9\) + \({(y-3)}^2\over 5\) = 1
Example 2 : For what value of k does the line y = x + k touches the ellipse \(9x^2 + 16y^2\) = 144.
Solution : \(\because\) Equation of ellipse is \(9x^2 + 16y^2\) = 144 or \(x^2\over 16\) + \({(y-3)}^2\over 9\) = 1
comparing this with \(x^2\over a^2\) + \(y^2\over b^2\) = 1 then we get \(a^2\) = 16 and \(b^2\) = 9
and comparing the line y = x + k with y = mx + c m = 1 and c = k
If the line y = x + k touches the ellipse \(9x^2 + 16y^2\) = 144, then \(c^2\) = \(a^2m^2 + b^2\)
\(\implies\) \(k^2\) = 16 \(\times\) \(1^2\) + 9 \(\implies\) \(k^2\) = 25
\(\therefore\) k = \(\pm\)5
Example 3 : Find the equation of the tangents to the ellipse \(3x^2+4y^2\) = 12 which are perpendicular to the line y + 2x = 4
Solution : Let m be the slope of the tangent, since the tangent is perpendicular to the line y + 2x = 4
\(\therefore\) mx – 2 = -1 \(\implies\) m = \(1\over 2\)
Since \(3x^2+4y^2\) = 12 or \(x^2\over 4\) + \(y^2\over 3\) = 1
Comparing this with \(x^2\over a^2\) + \(y^2\over b^2\) = 1
\(\therefore\) \(a^2\) = 4 and \(b^2\) = 3
So the equation of the tangent are y = \(1\over 2\)x \(\pm\) \(\sqrt{4\times {1\over 4} + 3}\)
\(\implies\) y = \(1\over 2\)x \(\pm\) 2 or x – 2y \(\pm\) 4 = 0
Practice these given ellipse examples to test your knowledge on concepts of ellipse.