Question

For $a>0,$ let the curves $C_1:y^2=ax$ and $C_2:x^2=ay$ intersect at origin $O$ and a point $P.$ Let the line $x=b\left(0<b<a\right)$ intersect the chord $OP$ and the $x$ -axis at points $Q$ and $R,$ respectively. If the line $x=b$ bisects the area bounded by the curves, $C_1$ and $C_2,$ and the area of $∆OQR=\frac{1}{2},$ then ‘ $a$ ’ satisfies the equation:

JEE Main 2020 (08 Jan Shift 1)

Options

• A: $x^6-6x^3+4=0$
• B: $x^6-12x^3+4=0$
• C: $x^6+6x^3-4=0$
• D: $x^6-12x^3-4=0$

Question

Let $\mathrm{P}$ be a point on the parabola, $y^2=12x$ and $\mathrm{N}$ be the foot of the perpendicular drawn from $P$ , on the axis of the parabola. A line is now drawn through the mid-point $M$ of $PN$, parallel to its axis which meets the parabola at $Q$. If the $y-$intercept of the line $\mathrm{NQ}$ is $\frac{4}{3},$ then :

JEE Main 2020 (03 Sep Shift 1)

Options

• A: $PN=4$
• B: $MQ=\frac{1}{3}$
• C: $MQ=\frac{1}{4}$
• D: $PN=3$

Question

Let $P$ be a variable point on the parabola $y=4x^2+1.$ Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $P$ to the line $y=x$ is:

JEE Main 2021 (20 Jul Shift 2)

Options

• A: $(3x-y{)}^2+(x-3y)+2=0$
• B: $2(3x-y{)}^2+(x-3y)+2=0$
• C: $(3x-y{)}^2+2(x-3y)+2=0$
• D: $2(x-3y{)}^2+(3x-y)+2=0$

Question

Let $x=2t,y=\frac{t^2}{3}$ be a conic. Let $S$ be the focus and $B$ be the point on the axis of the conic such that $SA\perp BA$, where $A$ is any point on the conic. If $k$ is the ordinate of the centroid of the $\Delta SAB$, then $\underset{t\to 1}{\lim }k$ is equal to

JEE Main 2022 (25 Jun Shift 1)

Options

• A: $\frac{17}{18}$
• B: $\frac{19}{18}$
• C: $\frac{11}{18}$
• D: $\frac{13}{18}$

Question

If the length of the latus rectum of a parabola, whose focus is $\left(a,a\right)$ and the tangent at its vertex is $x+y=a$, is $16$, then $\left|a\right|$ is equal to

JEE Main 2022 (27 Jul Shift 2)

Options

• A: $2\sqrt{2}$
• B: $2\sqrt{3}$
• C: $4\sqrt{2}$
• D: $4$

Question

Let $P_1$ be a parabola with vertex $\left(3,2\right)$ and focus $\left(4,4\right)$ and $P_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of $P_2$ is $x+2y=$ _____.

JEE Main 2022 (24 Jun Shift 2)

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Question

Let a tangent to the curve $y^2 = 24x$ meet the curve $xy = 2$ at the points $A$ and $B$. Then the mid-points of such line segments $AB$ lie on a parabola with the

JEE Main 2023 (24 Jan Shift 1)

Options

• A: directrix $4x=3$
• B: directrix $4x=-3$
• C: Length of latus rectum $\frac{3}{2}$
• D: Length of latus rectum $2$

Question

Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of the point $(7,1)$ on $C$ is $d$, then $12 d$ equals ______

JEE Main 2024 (06 Apr Shift 1)

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Question

Let $A, B$ and $C$ be three points on the parabola $y^2=6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left(\frac{A M \cdot B N}{C D}\right)^2$ is equal to _________

JEE Main 2024 (09 Apr Shift 2)

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Question

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2x^2+54$ at points $(x, y)$ and $(-x, y)$ where $\mathrm{y}>0$ is :

JEE Main 2024 (30 Jan Shift 1)

Options

• A: $88$
• B: $122$
• C: $92$
• D: $108$

Question

Consider the circle $C: x^2+y^2=4$ and the parabola $P: y^2=8 x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct lines passing through the point $(\alpha, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$, then $(2 q-p)^2$ is equal to ________

JEE Main 2024 (09 Apr Shift 2)

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Question

Let $P(\alpha ,\beta )$ be a point on the parabola $y^2=4x$. If $P$ also lies on the chord of the parabola $x^2=8y$ whose mid point is $\left(1,\frac{5}{4}\right)$, then $(\alpha -28)(\beta -8)$ is equal to _______.

JEE Main 2024 (29 Jan Shift 2)

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