A) \[2\sqrt{5}\]
B) \[~2\sqrt{7}\]
C) \[4\]
D) \[2\sqrt{2}\]
Correct Answer: B
Solution :
| [b] A line \[y=mx+c\] be a tangent to ellipse |
| \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], if \[{{c}^{2}}={{a}^{2}}{{m}^{2}}+{{b}^{2}}\] |
| Here, eq. of tangent is: \[4y\,=-\,3x+12\sqrt{2}\] |
| \[\therefore \,\,y=-\frac{3}{4}x+3\sqrt{2}\] |
| \[\therefore \,\,\,{{\left( 3\sqrt{2} \right)}^{2}}={{a}^{2}}.{{\left( -\frac{3}{4} \right)}^{2}}+9\] |
| \[\therefore \,\,{{a}^{2}}=9\times \frac{16}{9}=16\] |
| \[\therefore \,\,Eccentricity\,\,of\,\,ellipse\,=e=\sqrt{1-\frac{9}{16}}=\frac{\sqrt{7}}{4}\] |
| \[\therefore \,\,\,\,Distance\text{ }between\text{ }foci=2ac=2.4.\frac{\sqrt{7}}{4}\] |
| \[=2\sqrt{7}\] |
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