TS EAMCET · Maths · Ellipse

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

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 $E$

Verified Solution

Answer & Solution

Correct Answer

(D) $A$ lies inside $C,$ but outside $E$

Step-by-step Solution

Detailed explanation

Given the ellipse E≡4x2+9y2-36=0......................i the circle C≡x2+y2-9=0........................ii and two points Ax1,y1≡(1, 2), Bx2,y2≡(2, 1) Now, E1≡4x12+9y12-36=412+922-36=4>0 So A lies outside E.…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Given the ellipse E 4x2+9y2-36=0, the circle C x2+y2-9=0 and two points A(1, 2), B(2, 1), which of the following is correct?

Step-by-step Solution

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