3D Geometry

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JEE Main 2023 (Online) 29th January Evening ShiftQuestion 316

The plane $2x-y+z=4$ intersects the line segment joining the points $A(a,-2,4)$ and $B(2,b,-3)$ at the point $C$ in the ratio $2:1$ and the distance of the point $C$ from the origin is $\sqrt5$. If $ab<0$ and $P$ is the point $(a-b,b,2b-a)$ then $CP^{2}$ is equal to :

3D Geometry

$\frac{17}3$$\frac{16}3$$\frac{73}3$$\frac{97}3$

JEE Main 2023 (Online) 31st January Evening ShiftQuestion 317

The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the points $A,B,C$ is $(2,a,4)$, $a\in N$. If the volume of the tetrahedron $OABC$ is $144$ unit$^{3}$, then which of the following points is NOT on $P$?

3D Geometry

$(0,4,4)$$(3,0,4)$$(0,6,3)$$(2,2,4)$

JEE Main 2025 (Online) 29th January Evening ShiftQuestion 318

Let $P$ be the foot of the perpendicular from the point $(1,2,2)$ on the line $L:\frac{x-1}1=\frac{y+1}{-1}=\frac{z-2}2$. Let the line $\vec r=(-\hat i+\hat j-2\hat k)+\lambda(\hat i-\hat j+\hat k)$, $\lambda\in\mathbb R$, intersect the line $L$ at $Q$. Then $2(PQ)^{2}$ is equal to :

3D Geometry

$25$$19$$29$$27$

JEE Main 2023 (Online) 24th January Evening ShiftQuestion 319

Let the plane containing the line of intersection of the planes $P_1: x+(\lambda+4)y+z=1$ and $P_2: 2x+y+z=2$ pass through the points $(0,1,0)$ and $(1,0,1)$. Then the distance of the point $(2\lambda,\lambda,-\lambda)$ from the plane $P_2$ is

3D Geometry

$5\sqrt6$$4\sqrt6$$2\sqrt6$$3\sqrt6$

JEE Main 2023 (Online) 30th January Evening ShiftQuestion 320

A vector $\vec v$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at $45^{\circ}$ and to the $z$-axis at an acute angle. If a plane passing through the points $\left(\sqrt2,-1,1\right)$ and $(a,b,c)$, is normal to $\vec v$, then

3D Geometry

$\sqrt2a+b+c=1$$a+b+\sqrt2c=1$$a+\sqrt2b+c=1$$\sqrt2a-b+c=1$