Consider two straight lines, each of which is tangent to both the circle x 2 + y 2 = 1 2 and the parabola y…
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 ellipse is , then the which of the following statement(s) is (are) TRUE?
For the ellipse, the eccentricity is and the length of the latus rectum is 1
For the ellipse, the eccentricity is and the length of the latus rectum is
The area of the region bounded by the ellipse between the lines and is
The area of the region bounded by the ellipse between the lines and is
Solution
Let equation of common tangent is \(\mathrm{y}=\mathrm{m} \mathrm{x}+\frac{1}{\mathrm{~m}}\)
\(\therefore\left|\frac{0+0+\frac{1}{\mathrm{~m}}}{\sqrt{1+\mathrm{m}^{2}}}\right|=\frac{1}{\sqrt{2}} \Rightarrow \mathrm{m}^{4}+\mathrm{m}^{2}-2=0 \Rightarrow \mathrm{m}=\pm 1\) Equation of common tangents are \(\mathrm{y}=\mathrm{x}+1\) and \(\mathrm{y}=-\mathrm{x}+1\)
point \(Q\) is \((-1,0)\)
\(\therefore\) Equation of ellipse is \(\frac{x^{2}}{1}+\frac{y^{2}}{1 / 2}=1\)
(A) \(\mathrm{e}=\sqrt{1-\frac{1}{2}}=\frac{1}{\sqrt{2}}\) and \(\mathrm{LR}=\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=1\)
(C) Area \(2 \cdot \int_{1 / \sqrt{2}}^{1} \frac{1}{\sqrt{2}} \cdot \sqrt{1-\mathrm{x}^{2}} \mathrm{dx}=\sqrt{2}\left[\frac{\mathrm{x}}{2} \sqrt{1-\mathrm{x}^{2}}+\frac{1}{2} \sin ^{-1} \mathrm{x}\right]_{1 / \sqrt{2}}^{1}\) \(=\sqrt{2}\left[\frac{\pi}{4}-\left(\frac{1}{4}+\frac{\pi}{8}\right)\right]=\sqrt{2}\left(\frac{\pi}{8}-\frac{1}{4}\right]=\frac{\pi-2}{4 \sqrt{2}}\)
Correct answers are (A) and (C).
