A) \[c=\frac{2}{3}\]
B) \[c=\sqrt{\frac{73}{5}}\]
C) \[c=\frac{14}{\sqrt{73}}\]
D) \[c=\sqrt{\frac{5}{7}}\]
Correct Answer: C
Solution :
If the line y = mx + c is a normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\] then \[{{c}^{2}}=\frac{{{m}^{2}}{{\left( {{a}^{2}}-{{b}^{2}} \right)}^{2}}}{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}\] \[\left[ \text{Hence m = 2}\text{., }{{\text{a}}^{\text{2}}}\text{=9 and }{{\text{b}}^{\text{2}}}\text{=16} \right]\] \[=\frac{{{\left( 2 \right)}^{2}}{{\left( 9-16 \right)}^{2}}}{9+16\times {{\left( 2 \right)}^{2}}}\] \[=\frac{4\times 40}{9+64}=\frac{4\times 49}{73}=\frac{196}{73}\] \[\therefore \] \[c=\frac{14}{\sqrt{73}}\]
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