Equation of the ellipse whose axis are the axis of coordinates and which passes through the point (–3, 1) and has eccentricity \(\sqrt {\frac{2}{5}} \) is :
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\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)
\(\frac{9}{{{a^2}}} + \frac{1}{{{b^2}}} = 1\) ….(1)
case – 1 when a > b
b2 = a2 (1 – e2)
b2 = a2 (1 – 2/5)
5b2 = 3a2 ….(2)
from (1) & (2)
\(\frac{{9 \times 3}}{{5{b^2}}} + \frac{1}{{{b^2}}} = 1 \Rightarrow {b^2} = \frac{{32}}{5}\)
\ \({a^2} = \frac{{32}}{3}\)
\ \(\frac{{3{x^2}}}{{32}} + \frac{{5{y^2}}}{{32}} = 1\) Þ 3x2 + 5y2 – 32 = 0 Ans.
case – 2 when b > a
a2 = b2 (1 – e2)
\( = \frac{3}{5}{b^2}\) ….(3)
from (1) & (3)
\({a^2} = \frac{{48}}{5},{b^2} = 16\)
\ \(\frac{{5{x^2}}}{{48}} + \frac{{{y^2}}}{{16}} = 1\)
Þ 5x2 + 3y2 – 48 = 0