1
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

If the foci of a hyperbola are same as that of the ellipse $$\frac{x^2}{9}+\frac{y^2}{25}=1$$ and the eccentricity of the hyperbola is $$\frac{15}{8}$$ times the eccentricity of the ellipse, then the smaller focal distance of the point $$\left(\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right)$$ on the hyperbola, is equal to

A
$$14 \sqrt{\frac{2}{5}}-\frac{4}{3}$$
B
$$7 \sqrt{\frac{2}{5}}+\frac{8}{3}$$
C
$$7 \sqrt{\frac{2}{5}}-\frac{8}{3}$$
D
$$14 \sqrt{\frac{2}{5}}-\frac{16}{3}$$
2
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$P$$ be a point on the hyperbola $$H: \frac{x^2}{9}-\frac{y^2}{4}=1$$, in the first quadrant such that the area of triangle formed by $$P$$ and the two foci of $$H$$ is $$2 \sqrt{13}$$. Then, the square of the distance of $$P$$ from the origin is

A
26
B
22
C
20
D
18
3
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

Let $$e_1$$ be the eccentricity of the hyperbola $$\frac{x^2}{16}-\frac{y^2}{9}=1$$ and $$e_2$$ be the eccentricity of the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a} > \mathrm{b}$$, which passes through the foci of the hyperbola. If $$\mathrm{e}_1 \mathrm{e}_2=1$$, then the length of the chord of the ellipse parallel to the $$x$$-axis and passing through $$(0,2)$$ is :

A
$$\frac{8 \sqrt{5}}{3}$$
B
$$3 \sqrt{5}$$
C
$$4 \sqrt{5}$$
D
$$\frac{10 \sqrt{5}}{3}$$
4
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1

Let R be a rectangle given by the lines $$x=0, x=2, y=0$$ and $$y=5$$. Let A$$(\alpha,0)$$ and B$$(0,\beta),\alpha\in[0,2]$$ and $$\beta\in[0,5]$$, be such that the line segment AB divides the area of the rectangle R in the ratio 4 : 1. Then, the mid-point of AB lies on a :

A
hyperbola
B
straight line
C
parabola
D
circle
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