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$R$ and $r$ are the circum-radius and in-radius of a regular tetrahedron respectively in terms of the length $k$ of each edge. If $R^{2} + r^{2} = \frac{p}{q} k^{2},$ where $p, q \in I$ , then absolute minimum value of $p + q$ is

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Important Questions on Three Dimensional Geometry

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 15

A line

L

on the plane

2 x + y - 3 z + 5 = 0

is at a distance

3

unit from the point

P (1, 2, 3) .

A spider.starts moving from point

A

and after moving

4

units along the line

\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 3}{- 3}

it reaches to point

P

and from

P

, it jumps to line

L

along the shortest distance and then moves

12

units along the line

L

to reach at point

B .

The distance between points

A

and

B

k

units, then the value of

k

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 17

The direction cosines of the lines bisecting the angle between the lines whose direction cosines are

l_{1}, m_{1}, n_{1}

and

l_{2}, m_{2}, n_{2}

and the angle between these lines is

θ,

are

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 18

Aline passes through a point

A

with position vector

3 \hat{i} + \hat{j} - \hat{k}

and is parallel to the vector

2 \hat{i} - \hat{j} + 2 \hat{k} .

P

is a point on this line such that

A P = 15

units, then the position vector of the point

P

is/are

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 19

Acute angle between the lines

\frac{x - 1}{ℓ} = \frac{y + 1}{m} = \frac{z}{n}

and

\frac{x + 1}{m} = \frac{y - 3}{n} = \frac{z - 1}{ℓ}

where

ℓ > m > n,

and

ℓ, m, n

are the roots of the cubic equation

x^{3} + x^{2} - 4 x = 4

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 20

The line

\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 1}{- 1}

intersects the curve

x y = c^{2}, z = 0

C

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 21

Three distinct lines

\frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 3}{1}, \frac{x - 1}{5} = \frac{2 y - 4}{3} = \frac{3 z - 9}{1}, \frac{x - λ^{2}}{3} = \frac{y - 2}{2} = \frac{z - 3}{λ}

are concurrent the value of

λ

may be

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 22

The line

\frac{x + 6}{5} = \frac{y + 10}{3} = \frac{z + 14}{8}

is the hypotenuse of an isosceles right angle triangle whose opposite vertex is

(7, 2, 4)

Then the equation of remaining sides is/are -

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Alpha Question Bank for Engineering: Mathematics>Chapter 21 - Three Dimensional Geometry>Exercise-2>Q 23

Two lines are

L_{1} : \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 1}{2}; L_{2} : \frac{x - 1}{2} = \frac{y - 1}{1} = \frac{z - 1}{2}

. Equation of line passing through

(2, 1, 3)

and equally inclined to

L_{1} & L_{2}

is/are

R and r are the circum-radius and in-radius of a regular tetrahedron respectively in terms of the length k of each edge. If R2+r2=pqk2, where p, q∈I, then absolute minimum value of p+q is

Important Questions on Three Dimensional Geometry

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$R$ and $r$ are the circum-radius and in-radius of a regular tetrahedron respectively in terms of the length $k$ of each edge. If $R^{2} + r^{2} = \frac{p}{q} k^{2},$ where $p, q \in I$ , then absolute minimum value of $p + q$ is