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10-2. Parabola, Ellipse, Hyperbola
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જે ઉપવલયની અક્ષો યામાક્ષો હોય અને જે બિંદુ $(-3, 1)$માંથી પસાર થતું હોય અને ઉત્કેન્દ્રીતા $\sqrt {2/5} $ હોય, તે ઉપવલયનું સમીકરણ :
A
$3x^2 + 5y^2 - 15 = 0$
B
$5x^2 + 3y^2 - 32 =0$
C
$3x^2 + 5y^2 - 32 = 0$
D
$5x^2 + 3y^2 - 48 = 0$
Solution
Let the ellipse equation be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$e=\sqrt{\frac{2}{5}}$
$\text { or, } \sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{\frac{2}{5}}$
Squaring both sides,
$5\left( a ^2- b ^2\right)=2 a ^2$
$\text { or, } a ^2=\frac{5}{3} b ^2$
And the ellipse passes through the point $(-3,1)$
$\frac{9}{a^2}+\frac{1}{b^2}=1$
$9 b^2+a^2=a^2 b^2$
$9 b^2+\frac{5}{3} b^2=\frac{5}{3} b^4$
$\frac{32}{3}=\frac{5}{3} b^2$
$b^2=\frac{32}{5}$
$a^2=\frac{32}{3}$
$\text { Equation }: \frac{3 x^2}{32}+\frac{5 y^2}{32}=1$
$\Rightarrow 3 x^2+5 y^2=32$
Standard 11
Mathematics
