A) \[\frac{31}{10}\]
B) \[\frac{29}{10}\]
C) \[\frac{21}{10}\]
D) \[\frac{27}{10}\]
Correct Answer: D
Solution :
[d] Equation of the ellipse is \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{1}=1\] |
An end of the major axis A be say (3, 0) and an end of the minor axis B be say (0,1). Equations of AB is therefore.\[\frac{x}{3}+\frac{y}{1}=1\] ? (1) |
Equation of the auxiliary circle is \[{{x}^{2}}+{{y}^{2}}=9\] |
? (2) |
Solving the equation (1) and (2) we get |
\[{{x}^{2}}+{{\left( 1-\frac{x}{3} \right)}^{2}}=9\Rightarrow {{x}^{2}}+1+\frac{{{x}^{2}}}{9}-\frac{2x}{3}=9\] |
\[\Rightarrow 5{{x}^{2}}-3x-36=0\Rightarrow (5x+12)(x-3)=0\] |
\[\therefore x=-\frac{12}{5}\Rightarrow y=1-\frac{1}{3}\left( -\frac{12}{5} \right)=\frac{9}{5}\] |
\[\therefore \] Coordinates of M are \[\left( -\frac{12}{5},\frac{9}{5} \right)\] |
area of \[\Delta AOM=\frac{1}{2}.OA.MN=\frac{1}{2}\times 3\times \frac{9}{5}=\frac{27}{10}\] |
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