Question

Let $ABC$ be a triangle with $A(-3,1)$ and $\angle ACB=\theta ,0<\theta <\frac{\pi }{2}$. If the equation of the median through $\mathrm{B}$ is $2x+y-3=0$ and the equation of angle bisector of $\mathrm{C}$ is $7x-4\mathrm{y}-1=0$, then $\tan \theta$ is equal to:

JEE Main 2021 (26 Aug Shift 1)

Options

• A: $\frac{3}{4}$
• B: $\frac{4}{3}$
• C: $2$
• D: $\frac{1}{2}$

Question

Let $A\left(1,0\right),B\left(6,2\right)$ and $C\left(\frac{3}{2},6\right)$ be the vertices of a triangle $ABC.$ If $P$ is a point inside the triangle $ABC$ such that the triangles $APC,APB$ and $BPC$ have equal areas, then the length of the line segment $PQ,$ where $Q$ is the point $\left(-\frac{7}{6},-\frac{1}{3}\right),$ is

JEE Main 2020 (07 Jan Shift 1)

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Question

A triangle is formed by $X$-axis, $Y$-axis and the line $3x+4y=60$. Then the number of points $P\left(a,b\right)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is _____ .

JEE Main 2023 (25 Jan Shift 2)

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Question

Let $A\left(-2,-1\right), B\left(1,0\right), C\left(\alpha ,\beta \right)$ and $D\left(\gamma ,\delta \right)$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x-y=5$ and the point $D$ lies on $3x-2y=6$, then the value of $\left|\alpha +\beta +\gamma +\delta \right|$ is equal to ______.

JEE Main 2024 (31 Jan Shift 2)

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Question

Let $m_1,m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^2+11a+3\left( m_1^2+m_2^2\right)=220$. If one vertex of the square is $\left(10\left(\cos \alpha -\sin \alpha \right),10\left(\sin \alpha +\cos \alpha \right)\right)$, where $\alpha \in \left(0,\frac{\pi }{2}\right)$ and the equation of one diagonal is $\left(\mathrm{cos\alpha }-\mathrm{sin\alpha }\right)x+\left(\sin \alpha +\cos \alpha \right)y=10$, then $72\left({\sin }^4\alpha +{\cos }^4\alpha \right)+a^2-3a+13$ is equal to

JEE Main 2022 (29 Jul Shift 2)

Options

• A: $119$
• B: $128$
• C: $145$
• D: $155$

Question

A triangle $ABC$ lying in the first quadrant has two vertices as $A\left(1,2\right)$ and $B\left(3,1\right)$. If$\angle BAC={90}^{\mathrm{o}},$and $ar\left(\Delta \mathrm{ABC}\right)=5\sqrt{5}$ sq. units, then the abscissa of the vertex $\mathrm{C}$ is :

JEE Main 2020 (04 Sep Shift 1)

Options

• A: $1+\sqrt{5}$
• B: $1+2\sqrt{5}$ 
• C: $2+\sqrt{5}$
• D: $2\sqrt{5}-1$

Question

The equations of the sides $AB,BC$ and $CA$ of a triangle $ABC$ are $2x+y=0, x+py=39$ and $x-y=3$ respectively and $P\left(2,3\right)$ is its circumcentre. Then which of the following is NOT true

JEE Main 2022 (27 Jul Shift 2)

Options

• A: ${\left(AC\right)}^2=9p$
• B: ${\left(AC\right)}^2+p^2=136$
• C: $32<$area $\left(∆ABC\right)<36$
• D: $34<$ area $\left(∆ABC\right)<38$

Question

Let two straight lines drawn from the origin $\mathrm{O}$ intersect the line $3 x+4 y=12$ at the points $\mathrm{P}$ and $\mathrm{Q}$ such that $\triangle \mathrm{OPQ}$ is an isosceles triangle and $\angle \mathrm{POQ}=90^{\circ}$. If $l=\mathrm{OP}^2+\mathrm{PQ}^2+\mathrm{QO}^2$, then the greatest integer less than or equal to $l$ is :

JEE Main 2024 (05 Apr Shift 1)

Options

• A: 42
• B: 46
• C: 44
• D: 48

Question

Let $A\left(\frac{3}{\sqrt{a}},\sqrt{a}\right),a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D\left(3\cos \theta ,a\sin \theta \right)$,  is a point in the fourth quadrant such that the maximum area of $∆ACD$ is $12$ square units, then $a$ is equal to _____

JEE Main 2022 (24 Jun Shift 1)

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Question

A ray of light coming from the point $P(1,2)$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R(4,3)$. If the point $S(h, k)$ is such that PQRS is a parallelogram, then $h k^2$ is equal to :

JEE Main 2024 (09 Apr Shift 1)

Options

• A: 70
• B: 80
• C: 60
• D: 90

Question

Consider a triangle $\mathrm{ABC}$ having the vertices $\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$ and $\mathrm{C}(\gamma, \delta)$ and angles $\angle A B C=\frac{\pi}{6}$ and $\angle B A C=\frac{2 \pi}{3}$. If the points $\mathrm{B}$ and $\mathrm{C}$ lie on the line $y=x+4$, then $\alpha^2+\gamma^2$ is equal to ________

JEE Main 2024 (04 Apr Shift 2)

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Question

Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm{PAB}$ is 10 . If the locus of $\mathrm{P}$ is $\mathrm{a} x+\mathrm{b} y=15$, then $5 \mathrm{a}+2 \mathrm{~b}$ is :

JEE Main 2024 (05 Apr Shift 2)

Options

• A: 6
• B: $-\frac{6}{5}$
• C: 4
• D: $-\frac{12}{5}$

Question

Let $ABC$ be an isosceles triangle in which $A$ is at $\left(-1,0\right), \angle A=\frac{2\pi }{3}, AB=AC$ and $B$ is on the positive $x-$axis. If $BC=4\sqrt{3}$  and the line $BC$ intersects the line $y=x+3$ at $\left(\alpha ,\beta \right),$ then $\frac{{\beta }^4}{{\alpha }^2}$ is:

JEE Main 2024 (01 Feb Shift 2)

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Question

Let $A\left(-1,1\right), B\left(3,4\right)$ and $\mathrm{C}\left(2,0\right)$ be given three points. A line $y=mx, m>0$ , intersects lines $AC$ and $BC$ at point $P$ and $Q$ respectively. Let $A_1$ and $A_2$ be the areas of $\Delta ABC$ and $\Delta PQC$ respectively, such that $A_1=3A_2$, then the value of $m$ is equal to :

JEE Main 2021 (16 Mar Shift 2)

Options

• A: $\frac{4}{15}$
• B: $1$
• C: $2$
• D: $3$

Question

The vertices of a triangle are $\mathrm{A}(-1,3), \mathrm{B}(-2,2)$ and $\mathrm{C}(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

JEE Main 2024 (04 Apr Shift 1)

Options

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

Question

If the orthocentre of the triangle, whose vertices are $\left(1,2\right),\left(2,3\right)$ and $\left(3,1\right)$ is $\left(\alpha ,\beta \right)$, then the quadratic equation whose roots are $\alpha +4\beta$ and $4\alpha +\beta$, is

JEE Main 2023 (01 Feb Shift 1)

Options

• A: $x^2-19x+90=0$
• B: $x^2-18x+80=0$
• C: $x^2-22x+120=0$
• D: $x^2-20x+99=0$

Question

A ray of light passing through the point $P\left(2,3\right)$ reflects on the $X$-axis at point $A$ and the reflected ray passes through the point $Q\left(5,4\right)$. Let $R$ be the point that divides the line segment $AQ$ internally into the ratio $2:1$. Let the co-ordinates of the foot of the perpendicular $M$ from $R$ on the bisector of the angle $PAQ$ be $\left(\alpha ,\beta \right)$. Then, the value of $7\alpha +3\beta$ is equal to _____.

JEE Main 2022 (28 Jun Shift 1)

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Question

Let $A(0,1), B(1,1)$ and $C(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y^2=4ax$ passing through $D$ is $(\alpha +\beta \sqrt{2},0)$, where $\alpha$ and $\beta$ are rational numbers, then $\frac{\alpha }{{\beta }^2}$ is equal to

JEE Main 2023 (08 Apr Shift 2)

Options

• A: $8$
• B: $12$
• C: $6$
• D: $\frac{9}{2}$