1
JEE Advanced 2023 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?
A
The area of the quadrilateral $A_1 A_2 A_3 A_4$ is 35 square units
B
The area of the quadrilateral $A_1 A_2 A_3 A_4$ is 36 square units
C
The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$
D
The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$
2
JEE Advanced 2019 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :

$${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$

R1 : rectangle of largest area, with sides parallel to the axes, inscribed in E1;

En : ellipse $${{{x^2}} \over {a_n^2}} + {{{y^2}} \over {b_n^2}} = 1$$ of the largest area inscribed in $${R_{n - 1}},n > 1$$;

Rn : rectangle of largest area, with sides parallel to the axes, inscribed in En, n > 1.

Then which of the following options is/are correct?
A
The eccentricities of E18 and E19 are not equal.
B
The distance of a focus from the centre in E9 is $${{\sqrt 5 } \over {32}}$$.
C
$$\sum\limits_{n = 1}^N {(area\,of\,{R_n})}$$ < 24, for each positive integer N.
D
The length of latusrectum of E9 is $${1 \over 6}$$
3
JEE Advanced 2018 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is $$\sqrt 2$$, then which of the following statement(s) is (are) TRUE?
A
For the ellipse, the eccentricity is 1$$\sqrt 2$$ and the length of the latus rectum is 1
B
For the ellipse, the eccentricity is 1/2 and the length of the latus rectum is 1/2
C
The area of the region bounded by the ellipse between the lines $$x = {1 \over {\sqrt 2 }}$$ and x = 1 is $${1 \over {4\sqrt 2 }}(\pi - 2)$$
D
The area of the region bounded by the ellipse between the lines $$x = {1 \over {\sqrt 2 }}$$ and x = 1 is $${1 \over {16}}(\pi - 2)$$
4
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $${E_1}$$ and $${E_2}$$ be two ellipses whose centres are at the origin. The major axes of $${E_1}$$ and $${E_2}$$ lie along the $$x$$-axis and the $$y$$-axis, respectively. Let $$S$$ be the circle $${x^2} + {\left( {y - 1} \right)^2} = 2$$. The straight line $$x+y=3$$ touches the curves $$S$$, $${E_1}$$ and $${E_2}$$ at $$P, Q$$ and $$R$$ respectively. Suppose that $$PQ = PR = {{2\sqrt 2 } \over 3}$$. If $${e_1}$$ and $${e_2}$$ are the eccentricities of $${E_1}$$ and $${E_2}$$, respectively, then the correct expression(s) is (are)
A
$$\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 = {{43} \over {40}}$$
B
$${e_1}{e_2} = {{\sqrt 7 } \over {2\sqrt {10} }}$$
C
$$\left| {\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 } \right| = {5 \over 8}$$
D
$${e_1}{e_2} = {{\sqrt 3 } \over 4}$$
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