A) \[\frac{4}{5}\]
B) \[\frac{3}{5}\]
C) \[\frac{4}{15}\]
D) \[\frac{9}{5}\]
Correct Answer: A
Solution :
Given curve is \[25{{x}^{2}}+9{{y}^{2}}-150x-90y+225=0\] \[\Rightarrow \] \[(25{{x}^{2}}-150x)+(9{{y}^{2}}-90y)+225=0\] \[\Rightarrow \] \[25({{x}^{2}}-6x)+9({{y}^{2}}-10y)+225=0\] \[\Rightarrow \] \[25\{{{x}^{2}}-6x+9-9\}+9\{{{y}^{2}}-10y+25-25\}\] \[+225=0\] \[\Rightarrow \]\[25{{(x-3)}^{2}}-225+9{{(y-5)}^{2}}-225+225=0\] \[\Rightarrow \] \[25{{(x-3)}^{2}}+9{{(y-5)}^{2}}=225\] \[\Rightarrow \] \[\frac{{{(x-3)}^{2}}}{9}+\frac{{{(y-5)}^{2}}}{25}=1\] \[[\because \] \[(b>a)\]] Which represent an ellipse Whose eccentricity \[e=\sqrt{\frac{{{b}^{2}}-{{a}^{2}}}{{{b}^{2}}}}=\sqrt{\frac{25-9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}\]
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