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Question
Find the equation of the ellipse in the case:
The ellipse passes through (1, 4) and (−6, 1).
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Solution
\[ \text{ The ellipse passes through} \left( 1,4 \right)\text{ and} \left( - 6, 1 \right).\]
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
\[ \Rightarrow \frac{1}{a^2} + \frac{16}{b^2} = 1\]
\[\text{ Let} \frac{1}{a^2} = \alpha \text{ and } \frac{1}{b^2} = \beta\]
\[\text{ Then } \alpha + 16\beta = 1 . . (1)\]
\[\text{ It also passes through } \left( - 6, 1 \right).\]
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
\[ \Rightarrow \frac{36}{a^2} + \frac{1}{b^2} = 1\]
\[ \Rightarrow 36\alpha + \beta = 1 . . . (2)\]
\[\text{ Solving eqs. (1) and (2), we get } :\]
\[\alpha = \frac{3}{115} \text{ and } \beta = \frac{7}{115}\]
\[\text{ Substituting the values, we get } :\]
\[\frac{3 x^2}{115} + \frac{7 y^2}{115} = 1\]
\[ \Rightarrow \frac{3 x^2 + 7 y^2}{115} = 1\]
\[ \Rightarrow 3 x^2 + 7 y^2 = 115\]
\[\text{ This is the required equation of ellipse } .\]
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