Consider two straight lines, each of which is tangent to both the circle and the parabola Let these lines intersect at the point . Consider the ellipse whose center is at the origin and whose semi-major axis is . If the length of the minor axis of this

Question

Consider two straight lines, each of which is tangent to both the circle $x^{2} + y^{2} = \frac{1}{2}$ and the parabola $y^{2} = 4 x .$ Let these lines intersect at the point $Q$ . Consider the ellipse whose center is at the origin $O (0, 0)$ and whose semi-major axis is $O Q$ . If the length of the minor axis of this ellipse is $\sqrt{2}$ , then which of the following statement(s) is (are) TRUE? [2018]

Smart Achievers · Accepted Answer

(1, 3)

$Let the equation of common tangent is y = m x + \frac{1}{m}$

$∴$ $| \frac{0 + 0 + \frac{1}{m}}{\sqrt{1 + m^{2}}} |$ $\sqrt{2}$

$\Rightarrow m^{4} + m^{2} - 2 = 0 \Rightarrow m = \pm 1$

$∴$ $Equation of common tangents are$

$y = x + 1$ and $y = - x - 1$ $∴$ $Q \equiv (- 1, 0)$

$∴$ $Equation of ellipse is \frac{x^{2}}{1} + \frac{y^{2}}{1 / 2} = 1$

$(1)$ $e = \sqrt{1 - \frac{1}{2}} = \frac{1}{\sqrt{2}}$ $and latus rectum = \frac{2 b^{2}}{a} = \frac{2 {(\frac{1}{\sqrt{2}})}^{2}}{1} = 1$

$(3)$

$∴$ $Required area = 2 \int_{1 / \sqrt{2}}^{1} \frac{1}{\sqrt{2}} \sqrt{1 - x^{2}} d x$

$= \sqrt{2} {[\frac{x}{2} \sqrt{1 - x^{2}} + \frac{1}{2} \sin^{- 1} x]}_{1 / \sqrt{2}}^{1}$

$= \sqrt{2} [\frac{π}{4} - (\frac{1}{4} + \frac{π}{8})] = \sqrt{2} (\frac{π}{8} - \frac{1}{4}) = \frac{π - 2}{4 \sqrt{2}}$

Solution

1 For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is 1

2 For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$

3 The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{4 \sqrt{2}} (π - 2)$

4 The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{16} (π - 2)$

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Solution

1 For the ellipse, the eccentricity is 12 and the length of the latus rectum is 1

2 For the ellipse, the eccentricity is 12 and the length of the latus rectum is 12

3 The area of the region bounded by the ellipse between the lines x=12 and x=1 is 142(π-2)

4 The area of the region bounded by the ellipse between the lines x=12 and x=1 is 116(π-2)

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1 For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is 1

2 For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$

3 The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{4 \sqrt{2}} (π - 2)$

4 The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{16} (π - 2)$