Given the ellipse (E)4x^ 2 + 9y^ 2 - 36 = 0, ~the circle (C)x^ 2 + y^ 2 - 9 = 0 and two points A ( 1, 2 ) , B ( 2, 1 ) , & which of the following is correct?

Correct answer: D. A lies inside C, but outside E Given the ellipse E ≡ 4x^ 2 + 9y^ 2 - 36 = 0 . . . . . . . . . . . . . . . . . . . . . . (i) ~the circle C ≡ x^ 2 + y^ 2 - 9 = 0 . . . . . . . . . . . . . . . . . . . . . . . . (ii) and two points~A ( x _ 1 , y _ 1 )≡ ( 1, 2 ) , B ( x _ 2 , y _ 2 ) ≡ ( 2, 1 ) Now,~ E _ 1 ≡ 4 x1 ^ 2 + 9 y1 ^ 2 - 36 = 4( 1)2 + 9( 2)2 - 36 = 4 > 0 So~ A ~lies outside~ E . E _ 2 ≡ 4 x2 ^ 2 + 9 y2 ^ 2 - 36 = 4( 2)2 + 9( 1)2 - 36 = - 11 So~ B ~lies inside~ E . C _ 1 ≡ x1 ^ 2 + y1 ^ 2 - 9 = (1)2 + (2)2 - 9 = - 4 So~ A ~lies inside~ C . C _ 2 ≡ x2 ^ 2 + y2 ^ 2 - 9 = (2)2 + (1)2 - 9 = - 4 So~ B ~lies inside~ C . From the given options, option~ 4 & is correct.

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Ellipse

JEE Main · Mathematics · Ellipse

Given the ellipse (E)4x^ 2 + 9y^ 2 - 36 = 0, ~the circle (C)x^ 2 +...

Given the ellipse \left(E\right)4x^{2} + 9y^{2} - 36 = 0,

~the circle \left(C\right)x^{2} + y^{2} - 9 = 0

and two points

A ( 1, 2 ) , B ( 2, 1 ) ,

& which of the following is correct?

Options

A $B$ lies inside $C$ but outside $E$
B $B$ lies outside both $C$ and $E$
C $A$ lies inside both $C$ and $E$
D $A$ lies inside $C,$ but outside $E$

Correct answer

D. $A$ lies inside $C,$ but outside $E$

Step-by-step solution

Given the ellipse $E ≡ 4x^{2} + 9y^{2} - 36 = 0 . . . . . . . . . . . . . . . . . . . . . . \left(i\right)$

~the circle $C ≡ x^{2} + y^{2} - 9 = 0 . . . . . . . . . . . . . . . . . . . . . . . . \left(ii\right)$

and two points~A \left({x}_{1}, {y}_{1}\right)≡ ( 1, 2 ) , $B \left({x}_{2}, {y}_{2}\right)$ ≡ ( 2, 1 ) $Now,~$ {E}_{1} ≡ 4{x1}^{2} + 9{y1}^{2} - 36 = $4\left($ 1\right)2 + $9\left($ 2\right)2 - 36 = 4 > 0 $So~$ A $~lies outside~$ E . $$ {E}_{2} ≡ 4{x2}^{2} + 9{y2}^{2} - 36 = $4\left($ 2\right)2 + $9\left($ 1\right)2 - 36 = - 11 < 0 $So~$ B $~lies inside~$ E $.$ {C}_{1} ≡ {x1}^{2} + {y1}^{2} - 9 = \left(1\right)2 + \left(2\right)2 - 9 = - 4 < 0 $So~$ A $~lies inside~$ C $.$ {C}_{2} ≡ {x2}^{2} + {y2}^{2} - 9 = \left(2\right)2 + \left(1\right)2 - 9 = - 4 < 0 $So~$ B $~lies inside~$ C $.From the given options, option~$ 4 $& is correct.$

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