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Maths-

General

Easy

Question

An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is

$\sqrt{3}$
2
$2 \sqrt{2}$
$\sqrt{5}$

The correct answer is: 2

$table row cell 2 a equals 10 rightwards double arrow a equals 5 semicolon 2 b equals 8 rightwards double arrow b equals 4 end cell row cell c to the power of 2 end exponent equals 1 minus fraction numerator 16 over denominator 25 end fraction equals fraction numerator 9 over denominator 25 end fraction rightwards double arrow c equals fraction numerator 3 over denominator 5 end fraction end cell end table$

Focus = (3, 0)
Let the circle touches the ellipse at P and Q. Consider a tangent (to both circle and ellipse) at P. Let F(one focus) be the centre of the circle and other focus be G. A ray from F to P must retrace its path (normal to the circle). But the reflection propety the ray FP must be reflected along PG. This is possible only if P, F and G are collinear. Thus P must be the end of the major axis. Hence r = a – ae = 5 – 3 = 2 alternately normal to an ellipse at P must pass through the centre (3, 0) of the circle
$table row cell fraction numerator a x over denominator cos invisible function application theta end fraction minus fraction numerator b y over denominator sin invisible function application theta end fraction equals a to the power of 2 end exponent minus b to the power of 2 end exponent rightwards double arrow fraction numerator 5 x over denominator cos invisible function application theta end fraction minus fraction numerator 4 y over denominator sin invisible function application theta end fraction equals 9 open parentheses theta not equal to 0 text or end text fraction numerator pi over denominator 2 end fraction close parentheses end cell row cell fraction numerator 15 over denominator cos invisible function application theta end fraction minus 0 equals 9 rightwards double arrow c o s invisible function application theta equals fraction numerator 15 over denominator 9 end fraction rightwards double arrow text end text text w end text text h end text text i end text text c end text text h end text text end text text i end text text s end text text end text text n end text text o end text text t end text text end text text p end text text o end text text s end text text s end text text i end text text b end text text l end text text e end text text end text rightwards double arrow theta equals 0 text end text text o end text text r end text text end text pi divided by 2 end cell row cell text end text text b end text text u end text text t end text text end text theta not equal to pi divided by 2 rightwards double arrow theta equals 0 end cell row cell text end text text H end text text e end text text n end text text c end text text e end text text end text P equals left parenthesis 50 right parenthesis text end text text i end text text. end text text e end text text. end text text end text text e end text text n end text text d end text text end text text o end text text f end text text end text text m end text text a end text text j end text text o end text text r end text text end text text a end text text x end text text i end text text s end text text end text right square bracket end cell end table$

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