1
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1

Consider ellipses $$\mathrm{E}_{k}: k x^{2}+k^{2} y^{2}=1, k=1,2, \ldots, 20$$. Let $$\mathrm{C}_{k}$$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $$\mathrm{E}_{k}$$. If $$r_{k}$$ is the radius of the circle $$\mathrm{C}_{k}$$, then the value of $$\sum_\limits{k=1}^{20} \frac{1}{r_{k}^{2}}$$ is :

A
2870
B
3210
C
3320
D
3080
2
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1
Out of Syllabus

Let a circle of radius 4 be concentric to the ellipse $$15 x^{2}+19 y^{2}=285$$. Then the common tangents are inclined to the minor axis of the ellipse at the angle :

A
$$\frac{\pi}{4}$$
B
$$\frac{\pi}{3}$$
C
$$\frac{\pi}{6}$$
D
$$\frac{\pi}{12}$$
3
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
4
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
$$\frac{\sqrt{117}}{12}$$
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