1
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1

Let the ellipse $$E:{x^2} + 9{y^2} = 9$$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is $${m \over n}$$, where m and n are coprime, then $$m - n$$ is equal to :

A
15
B
16
C
17
D
18
2
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $$\alpha$$ and the number of persons who speak only Hindi is $$\beta$$, then the eccentricity of the ellipse $$25\left(\beta^{2} x^{2}+\alpha^{2} y^{2}\right)=\alpha^{2} \beta^{2}$$ is :

A
$$\frac{\sqrt{129}}{12}$$
B
$$\frac{3 \sqrt{15}}{12}$$
C
$$\frac{\sqrt{119}}{12}$$
D
$$\frac{\sqrt{117}}{12}$$
3
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1
Out of Syllabus

If the maximum distance of normal to the ellipse $$\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b < 2$$, from the origin is 1, then the eccentricity of the ellipse is :

A
$$\frac{\sqrt{3}}{4}$$
B
$$\frac{1}{2}$$
C
$$\frac{1}{\sqrt{2}}$$
D
$$\frac{\sqrt{3}}{2}$$
4
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

Let a line L pass through the point of intersection of the lines $$b x+10 y-8=0$$ and $$2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}$$. If the line $$\mathrm{L}$$ also passes through the point $$(1,1)$$ and touches the circle $$17\left(x^{2}+y^{2}\right)=16$$, then the eccentricity of the ellipse $$\frac{x^{2}}{5}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$$ is :

A
$$\frac{2}{\sqrt{5}}$$
B
$$\sqrt{\frac{3}{5}}$$
C
$$\frac{1}{\sqrt{5}}$$
D
$$\sqrt{\frac{2}{5}}$$
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