A) \[{{x}^{2}}\text{cose}{{\text{c}}^{2}}\theta -{{y}^{2}}{{\sec }^{2}}\theta =1\]
B) \[{{x}^{2}}{{\sec }^{2}}\theta -{{y}^{2}}\text{cose}{{\text{c}}^{2}}\theta =1\]
C) \[{{x}^{2}}{{\sin }^{2}}\theta -{{y}^{2}}{{\cos }^{2}}\theta =1\]
D) \[{{x}^{2}}{{\cos }^{2}}\theta -{{y}^{2}}{{\sin }^{2}}\theta =1\]
Correct Answer: A
Solution :
[a]| Equation of ellipse |
| \[3{{x}^{2}}+4{{y}^{2}}=12\] |
| \[\Rightarrow \]\[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{3}=1\] |
| Focus of ellipse \[(\pm 1,0).\] |
| Focus of ellipse is also focus of hyperbola. |
| \[\therefore \] \[ae=1\] |
| Length of transverse axis \[(2a)=2\sin \theta \] |
| \[\therefore \] \[a=\sin \theta \] |
| In hyperbola \[{{(ae)}^{2}}={{a}^{2}}+{{b}^{2}}\] |
| \[1={{\sin }^{2}}\theta +{{b}^{2}}\] |
| \[\Rightarrow \] \[{{b}^{2}}={{\cos }^{2}}\theta \] |
| Equation of hyperbola \[\frac{{{x}^{2}}}{{{\sin }^{2}}\theta }-\frac{{{y}^{2}}}{{{\cos }^{2}}\theta }=1\] |
| \[\Rightarrow \]\[{{x}^{2}}\text{cose}{{\text{c}}^{2}}\theta -{{y}^{2}}{{\sec }^{2}}\theta =1\] |
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