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The ellipse x2 + 4y2 = 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is : [2009]
- a)x2 + 12y2= 16
- b)4 x2 + 48y2= 48
- c)4 x2 + 64y2 = 48
- d)x2 + 16y2= 16
Correct answer is option 'A'. Can you explain this answer?
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The ellipse x2 + 4y2= 4 is inscribed in a rectangle aligned with the c...
Solution:
Given that the ellipse x^2/4 + y^2 = 1 is inscribed in a rectangle aligned with the coordinate axes and this rectangle is inscribed in another ellipse that passes through the point (4, 0).
Let's find the equation of the outer ellipse.
Step 1: Find the coordinates of the four corners of the rectangle.
The corners of the rectangle can be obtained by solving the equation of the ellipse with x = ±2 and y = ±1.
For x = 2, we have 4/4 + y^2 = 1, which gives y = ±√3.
So, the coordinates are (2, √3) and (2, -√3).
Similarly, for x = -2, we have 4/4 + y^2 = 1, which gives y = ±√3.
So, the coordinates are (-2, √3) and (-2, -√3).
Step 2: Find the equation of the ellipse passing through the points (2, √3), (2, -√3), (-2, √3), and (-2, -√3).
Using the standard form of an ellipse, the equation is:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1, where (h, k) is the center of the ellipse, and a and b are the semi-major and semi-minor axes, respectively.
Step 3: Find the center of the ellipse.
The center of the ellipse is the midpoint of the diagonals of the rectangle.
The midpoint of the diagonals is ((2 + (-2))/2, (√3 + (-√3))/2) = (0, 0).
Step 4: Find the semi-major and semi-minor axes.
The semi-major axis is the distance from the center to one of the corners of the rectangle, which is 2.
The semi-minor axis is the distance from the center to one of the sides of the rectangle, which is 1.
Therefore, the equation of the outer ellipse is:
(x - 0)^2/2^2 + (y - 0)^2/1^2 = 1
Simplifying, we get x^2/4 + y^2 = 1.
Hence, the correct answer is option A) x^2 - 12y^2 = 16.
Given that the ellipse x^2/4 + y^2 = 1 is inscribed in a rectangle aligned with the coordinate axes and this rectangle is inscribed in another ellipse that passes through the point (4, 0).
Let's find the equation of the outer ellipse.
Step 1: Find the coordinates of the four corners of the rectangle.
The corners of the rectangle can be obtained by solving the equation of the ellipse with x = ±2 and y = ±1.
For x = 2, we have 4/4 + y^2 = 1, which gives y = ±√3.
So, the coordinates are (2, √3) and (2, -√3).
Similarly, for x = -2, we have 4/4 + y^2 = 1, which gives y = ±√3.
So, the coordinates are (-2, √3) and (-2, -√3).
Step 2: Find the equation of the ellipse passing through the points (2, √3), (2, -√3), (-2, √3), and (-2, -√3).
Using the standard form of an ellipse, the equation is:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1, where (h, k) is the center of the ellipse, and a and b are the semi-major and semi-minor axes, respectively.
Step 3: Find the center of the ellipse.
The center of the ellipse is the midpoint of the diagonals of the rectangle.
The midpoint of the diagonals is ((2 + (-2))/2, (√3 + (-√3))/2) = (0, 0).
Step 4: Find the semi-major and semi-minor axes.
The semi-major axis is the distance from the center to one of the corners of the rectangle, which is 2.
The semi-minor axis is the distance from the center to one of the sides of the rectangle, which is 1.
Therefore, the equation of the outer ellipse is:
(x - 0)^2/2^2 + (y - 0)^2/1^2 = 1
Simplifying, we get x^2/4 + y^2 = 1.
Hence, the correct answer is option A) x^2 - 12y^2 = 16.
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The ellipse x2 + 4y2= 4 is inscribed in a rectangle aligned with the c...
The given ellipse is 
So A = (2, 0) and B= (0, 1) If PQRS is the rectangle in which it is inscribed, then P = (2, 1).
be the ellipse circumscribing the rectangle PQRS.
Then it passes through P (2,1 )
Also, given that, it passes through (4, 0)
⇒ b2 = 4/3 [substituting a2 = 16 in eqn (a)]
∴ The required ellipse is 
or x2 + 12y2 =16
or x2 + 12y2 =16Attention JEE Students!
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The ellipse x2 + 4y2= 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is : [2009]a)x2 + 12y2= 16b)4 x2 + 48y2= 48c)4 x2 + 64y2= 48d)x2 + 16y2= 16Correct answer is option 'A'. Can you explain this answer?
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