1
JEE Main 2023 (Online) 29th January Morning Shift
Numerical
+4
-1

Let the equation of the plane P containing the line $$x+10=\frac{8-y}{2}=z$$ be $$ax+by+3z=2(a+b)$$ and the distance of the plane $$P$$ from the point (1, 27, 7) be $$c$$. Then $$a^2+b^2+c^2$$ is equal to __________.

2
JEE Main 2023 (Online) 29th January Morning Shift
Numerical
+4
-1

Let the co-ordinates of one vertex of $$\Delta ABC$$ be $$A(0,2,\alpha)$$ and the other two vertices lie on the line $${{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\Delta ABC$$ is 21 sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ___________.

3
JEE Main 2023 (Online) 25th January Evening Shift
Numerical
+4
-1

If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line $${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}$$ is $$\alpha$$, then 28$$\alpha^2$$ is equal to ____________.

4
JEE Main 2023 (Online) 25th January Morning Shift
Numerical
+4
-1

Let the equation of the plane passing through the line $$x - 2y - z - 5 = 0 = x + y + 3z - 5$$ and parallel to the line $$x + y + 2z - 7 = 0 = 2x + 3y + z - 2$$ be $$ax + by + cz = 65$$. Then the distance of the point (a, b, c) from the plane $$2x + 2y - z + 16 = 0$$ is ____________.

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