InterviewSolution
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InterviewSolution
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An ellipse E,(x^(2))/(a^(2))+(y^(2))/(b^(2))=1, centred at point O has AB and CD as its major and minor axes, respectively. Let S_(1) be one of thefoci of the ellipse, the radius of the incircle of traingle OCS_(1) be 1 unit, adn OS_(1)=6 units The equation ofthe director circle of E is |
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Answer» `x^(2)+y^(2)=48.5`  `:.OS_(1)=ae=6,OC=b` Also,` CS_(1)=a` `:. "Area of " DeltaOCS_(1)=(1)/(2)=(OS_(1))xx(OC)=3b` `:.`Semi-perimeter of `DeltaOCS_(1)=(1)/(2)=(OS_(1)+OC+CS_(1))` `=(1)/(2)(6+a+b) ""(1)` `:.` In radius of `DeltaOCS_(1) ","( "Using" r=(Delta)/(S))` =` (3b)/((1)/(2)(6+a+b))=1""("Using"r=(Delta)/(S))` `or 5b=6+a""(2)` Also, `b^(2)=a^(2)-a^(2)e^(2)=a^(2)-36""(3)` From (2), we get `25(a^(2)-36)=36+a^(2)+12a` `or 2a^(2)-a-78=0` `a=(13)/(2),-6` `:. a=(13)/(2) and b=(5)/(2)` Area or ellipse `=piab=(65pi)/(4)` sq. UNIT Perimeter of `DeltaOCS_(1)=6+a+b=6+(13)/(2)+(5)/(2)=15` UNITS The equation of DIRECTOR circle is `x^(2)+y^(2)=a^(2)+b^(2)` or `x^(2)+y^(2)=(97)/(2)` |
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