Question

Find the equation of the ellipse in the case:

 The ellipse passes through (1, 4) and (−6, 1).

Answer 1

\[ \text{ The ellipse passes through} \left( 1,4 \right)\text{ and} \left( - 6, 1 \right).\]

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

\[ \Rightarrow \frac{1}{a^2} + \frac{16}{b^2} = 1\]

\[\text{ Let} \frac{1}{a^2} = \alpha \text{ and } \frac{1}{b^2} = \beta\]

\[\text{ Then } \alpha + 16\beta = 1 . . (1)\]

\[\text{ It also passes through } \left( - 6, 1 \right).\]

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

\[ \Rightarrow \frac{36}{a^2} + \frac{1}{b^2} = 1\]

\[ \Rightarrow 36\alpha + \beta = 1 . . . (2)\]

\[\text{ Solving eqs. (1) and (2), we get } :\]

\[\alpha = \frac{3}{115} \text{ and  } \beta = \frac{7}{115}\]

\[\text{ Substituting the values, we get } :\]

\[\frac{3 x^2}{115} + \frac{7 y^2}{115} = 1\]

\[ \Rightarrow \frac{3 x^2 + 7 y^2}{115} = 1\]

\[ \Rightarrow 3 x^2 + 7 y^2 = 115\]

\[\text{ This is the required equation of ellipse } .\]