| Statement 1: The circles \[{{x}^{2}}+{{y}^{2}}=9\] and \[(2x-3)(x-1)+2y(y-\sqrt{6})=0\] touches each other internally. |
| Statement 2: Circle described on the focal distance as diameter of the ellipse \[8{{x}^{2}}+9{{y}^{2}}=72\] touches the auxiliary circle \[{{x}^{2}}+{{y}^{2}}=9\] internally. |
A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.
B) Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.
C) Statement-1 is true, Statement-2 is false
D) Statement-1 is false, Statement-2 is true
Correct Answer: A
Solution :
Equation of ellipse in \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{8}=1\] Focus \[\equiv (1,\,0),\,e=\frac{1}{3}\] Also, a point on given ellipse is \[\left( 3\cos \theta ,\,\,2\sqrt{2}\,\sin \theta \right)\] Put \[\theta =\frac{\pi }{3}\], we get a point on given ellipse as \[\left( \frac{3}{2},\,\sqrt{6} \right)\]. Now, circle described on the focal distance as diameter of the ellipse is \[(2x-3)\,(x-1)\,+2y\left( y-\sqrt{6} \right)=0\] Which is true. (Using property of ellipse.)
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