Let the line intersect the ellipse at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is: [2026]

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Q.
Let the line $y - x = 1$ intersect the ellipse $\frac{x^{2}}{2} + \frac{y^{2}}{1} = 1$ at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is: [2026]

1 $π - \tan^{- 1} (\frac{1}{4})$

2 $\frac{π}{2} + \tan^{- 1} (\frac{1}{4})$

3 $\frac{π}{2} + 2 \tan^{- 1} (\frac{1}{4})$

4 $\frac{π}{2} - \tan^{- 1} (\frac{1}{4})$

Ans.
(2)

$By solving line & equation of ellipse we get x = 0$

$& x = - \frac{4}{3}$

$∴$ $B (- \frac{4}{3}, - \frac{1}{3})$

$m_{O B} = \tan θ = \frac{1}{4}$

$∵$ $∠ A O B = \frac{π}{2} + θ = \frac{π}{2} + \tan^{- 1} (\frac{1}{4})$

Solution

Q.
Let the line $y - x = 1$ intersect the ellipse $\frac{x^{2}}{2} + \frac{y^{2}}{1} = 1$ at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is: [2026]

1 $π - \tan^{- 1} (\frac{1}{4})$

2 $\frac{π}{2} + \tan^{- 1} (\frac{1}{4})$

3 $\frac{π}{2} + 2 \tan^{- 1} (\frac{1}{4})$

4 $\frac{π}{2} - \tan^{- 1} (\frac{1}{4})$

Ans.
(2)

$By solving line & equation of ellipse we get x = 0$

$& x = - \frac{4}{3}$

$∴$ $B (- \frac{4}{3}, - \frac{1}{3})$

$m_{O B} = \tan θ = \frac{1}{4}$

$∵$ $∠ A O B = \frac{π}{2} + θ = \frac{π}{2} + \tan^{- 1} (\frac{1}{4})$

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Solution

Q. Let the line y-x=1 intersect the ellipse x22+y21=1 at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is: [2026]

1 π-tan-1(14)

2 π2+tan-1(14)

3 π2+2tan-1(14)

4 π2-tan-1(14)

Ans. (2) By solving line & equation of ellipse we get x=0 & x=-43 ∴ B(-43,-13) mOB=tanθ=14 ∵ ∠AOB=π2+θ=π2+tan-1(14)

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Q.
Let the line $y - x = 1$ intersect the ellipse $\frac{x^{2}}{2} + \frac{y^{2}}{1} = 1$ at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is: [2026]

1 $π - \tan^{- 1} (\frac{1}{4})$

2 $\frac{π}{2} + \tan^{- 1} (\frac{1}{4})$

3 $\frac{π}{2} + 2 \tan^{- 1} (\frac{1}{4})$

4 $\frac{π}{2} - \tan^{- 1} (\frac{1}{4})$

Ans.
(2)

$By solving line & equation of ellipse we get x = 0$

$& x = - \frac{4}{3}$

$∴$ $B (- \frac{4}{3}, - \frac{1}{3})$

$m_{O B} = \tan θ = \frac{1}{4}$

$∵$ $∠ A O B = \frac{π}{2} + θ = \frac{π}{2} + \tan^{- 1} (\frac{1}{4})$