1
JEE Advanced 2023 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language
Let $P$ be a point on the parabola $y^2=4 a x$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $P F Q$, where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is
A
$(2,3)$
B
$(1,3)$
C
$(2,4)$
D
$(3,4)$
2
JEE Advanced 2022 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

Consider the ellipse

$$$ \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 $$$

Let $H(\alpha, 0), 0<\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.

List-I List-II
(I) If $\phi=\frac{\pi}{4}$, then the area of the triangle $F G H$ is (P) $\frac{(\sqrt{3}-1)^{4}}{8}$
(II) If $\phi=\frac{\pi}{3}$, then the area of the triangle $F G H$ is (Q) 1
(III) If $\phi=\frac{\pi}{6}$, then the area of the triangle $F G H$ is (R) $\frac{3}{4}$
(IV) If $\phi=\frac{\pi}{12}$, then the area of the triangle $F G H$ is (S) $\frac{1}{2 \sqrt{3}}$
(T) $\frac{3 \sqrt{3}}{2}$

The correct option is:

A
$(\mathrm{I}) \rightarrow(\mathrm{R}) ;(\mathrm{II}) \rightarrow(\mathrm{S}) ;(\mathrm{III}) \rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$
B
(I) $\rightarrow$ (R); (II) $\rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$
C
(I) $\rightarrow(\mathrm{Q}) ;(\mathrm{II}) \rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$
D
(I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (P)
3
JEE Advanced 2020 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-1
Change Language
Let a, b and $$\lambda $$ be positive real numbers. Suppose P is an end point of the latus return of the
parabola y2 = 4$$\lambda $$x, and suppose the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is
A
$${1 \over {\sqrt 2 }}$$
B
$${{1 \over 2}}$$
C
$${{1 \over 3}}$$
D
$${{2 \over 5}}$$
4
JEE Advanced 2019 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
Change Language
Let the circles

C1 : x2 + y2 = 9 and C2 : (x $$-$$ 3)2 + (y $$-$$ 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x $$-$$ h)2 + (y $$-$$ k)2 = r2 satisfies the following conditions :

(i) Centre of C3 is collinear with the centres of C1 and C2.

(ii) C1 and C2 both lie inside C3 and

(iii) C3 touches C1 at M and C2 at N.

Let the line through X and Y intersect C3 at Z and W, and let a common tangent of C1 and C3 be a tangent to the parabola x2 = 8$$\alpha $$y.

There are some expression given in the List-I whose values are given in List-II below.

JEE Advanced 2019 Paper 2 Offline Mathematics - Conic Sections Question 23 English

Which of the following is the only INCORRECT combination?
A
(III), (R)
B
(IV), (S)
C
(I), (P)
D
(IV), (U)
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