JEE Mathematics / Practice

3D Geometry

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JEE Main 2023 (Online) 12th April Morning ShiftQuestion 181

Let the plane $x+3y-2z+6=0$ meet the co-ordinate axes at the points $A,B,C$. If the orthocenter of the triangle $ABC$ is $\left(\alpha,\beta,\frac{6}{7}\right)$, then $98(\alpha+\beta)^{2}$ is equal to $\_\_\_\_\_\_$ .

3D Geometry

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JEE Main 2022 (Online) 26th July Evening ShiftQuestion 182

The largest value of $a$, for which the perpendicular distance of the plane containing the lines $\vec r=(\hat i+\hat j)+\lambda(\hat i+a\hat j-\hat k)$ and $\vec r=(\hat i+\hat j)+\mu(-\hat i+\hat j-a\hat k)$ from the point $(2,1,4)$ is $\sqrt3$, is ________.

3D Geometry

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JEE Main 2022 (Online) 25th June Morning ShiftQuestion 183

Let the lines $L_1:\vec{r}=\lambda\left(\hat{i}+2\hat{j}+3\hat{k}\right)$, $\lambda\in\mathbf{R}$, $L_2:\vec{r}=\left(\hat{i}+3\hat{j}+\hat{k}\right)+\mu\left(\hat{i}+\hat{j}+5\hat{k}\right)$; $\mu\in\mathbf{R}$, intersect at the point $S$. If a plane $ax+by-z+d=0$ passes through $S$ and is parallel to both the lines $L_1$ and $L_2$, then the value of $a+b+d$ is equal to $\_\_\_\_\_\_$ .

3D Geometry

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JEE Main 2023 (Online) 15th April Morning ShiftQuestion 184

Let the plane $P$ contain the line $2x+y-z-3=0=5x-3y+4z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point $A(8,-1,-19)$ from the plane $P$ measured parallel to the line $\frac{x}{-3}=\frac{y-5}{4}=\frac{2-z}{-12}$ is equal to $\_\_\_\_\_\_$ .

3D Geometry

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JEE Main 2024 (Online) 31st January Evening ShiftQuestion 185

A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a,b,c)$, where $a,b,c$ are non-negative integers, on the line $AB$ lies at a distance of $21$ units, from the point $A$. The distance between the points $P(a,b,c)$ and $Q(4,-12,3)$ is equal to $\_\_\_\_\_\_$ .

3D Geometry

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