JEE Main 2022 (27 June Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Three Dimensional Geometry: If two straight lines whose direction cosines are given by the relations $l + m - n = 0, 3 l^{2} + m^{2} + c n l = 0$ are parallel, then the positive value of $c$ is :

Q: Why do we divide by $m^2$ in the equation $(3 + c)l^2 + clm + m^2 = 0$?

Great question! We divide by $m^2$ to transform the homogeneous equation into a standard quadratic equation in terms of the ratio $l/m$. This allows us to treat the ratio as a single variable, making it much easier to apply the condition for equal roots.

Q: What if $m = 0$ when we divide by $m^2$?

That is a sharp observation! If $m = 0$, then from $l + m - n = 0$, we get $l = n$. Substituting this into $3l^2 + m^2 + cnl = 0$ gives $3l^2 + c(l)(l) = 0$, or $l^2(3 + c) = 0$. Since $l$ cannot be zero for a valid direction cosine, this would imply $c = -3$. However, the problem asks for the positive value of $c$, and our discriminant method covers the general case where $m$ is non-zero.

Select Answer:

Visualized Solution

Given Relations

Direction cosines $(l, m, n)$ satisfy:
$l + m - n = 0$
$3 l^{2} + m^{2} + c n l = 0$

Isolating $n$

From the linear equation:
$l + m - n = 0$
$⟹ n = l + m$

Substitution

Substitute $n = l + m$ into the quadratic equation:
$3 l^{2} + m^{2} + c (l + m) l = 0$

Expansion

Expand the substituted term:
$3 l^{2} + m^{2} + c l^{2} + c l m = 0$

Grouping Terms

Group the $l^{2}$ terms together:
$(3 + c) l^{2} + c l m + m^{2} = 0$

Forming a Quadratic in $\frac{l}{m}$

Divide the entire equation by $m^{2}$ :
$(3 + c) (\frac{l}{m})^{2} + c (\frac{l}{m}) + 1 = 0$

Parallel Lines Condition

For parallel lines, direction cosines are proportional.
The ratio $\frac{l}{m}$ must have a unique value.
Therefore, the quadratic equation must have equal roots.

Discriminant $D = 0$

For equal roots, Discriminant $D = b^{2} - 4 a c = 0$
Here, $a = 3 + c$ , $b = c$ , $c = 1$
$c^{2} - 4 (3 + c) (1) = 0$

Solving for $c$

Expand and simplify:
$c^{2} - 4 c - 12 = 0$
Factorizing the quadratic:
$(c - 6) (c + 2) = 0$

Final Answer

Possible values: $c = 6$ or $c = - 2$
The question asks for the positive value.
Final Answer: $c = 6$

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The Sigma Insight: Direction Cosines and Direction Ratios

Solution Diagram

Analyzing the Setup

In a 3D coordinate system, two parallel lines share the same direction cosines

(l, m, n)

. These represent a unit vector, implying the constraint

l^{2} + m^{2} + n^{2} = 1

We are provided with two specific relations governing these direction cosines: 1.

l + m - n = 0

3 l^{2} + m^{2} + c n l = 0

Our objective is to determine the positive value of the constant

c

that satisfies these conditions.

The Art of Reduction

We begin by simplifying the system using the linear relation

l + m - n = 0

. By isolating

n

, we obtain:

n = l + m

Next, we substitute this expression for

n

into the second equation,

3 l^{2} + m^{2} + c n l = 0

. This substitution yields:

3 l^{2} + m^{2} + c (l + m) l = 0

The Transformation

Expanding the terms of the equation, we get:

3 l^{2} + m^{2} + c l^{2} + c l m = 0

Grouping the terms by the powers of

l

and

m

, we arrive at a homogeneous quadratic equation:

(3 + c) l^{2} + c l m + m^{2} = 0

To solve for the ratio of the direction cosines, we divide the entire equation by

m^{2}

. Letting

x = \frac{l}{m}

, we obtain the standard quadratic form:

(3 + c) x^{2} + c x + 1 = 0

The Final Insight

For the lines to possess a unique direction, the ratio

x = \frac{l}{m}

must be unique. In algebraic terms, this requires the quadratic equation to have equal roots, which occurs if and only if the discriminant

D

is zero.

The discriminant

D = b^{2} - 4 a c

for our equation is:

D = c^{2} - 4 (3 + c) (1) = 0

Expanding and simplifying this expression leads to:

c^{2} - 4 c - 12 = 0

Factoring the quadratic equation, we find:

(c - 6) (c + 2) = 0

This yields two potential values:

c = 6

and

c = - 2

. Since the problem explicitly requires the positive value, we conclude that $c = 6$ .

Common Traps & FAQs

Why do we divide by

m^{2}

in the equation

(3 + c) l^{2} + c l m + m^{2} = 0

Great question! We divide by

m^{2}

to transform the homogeneous equation into a standard quadratic equation in terms of the ratio

l / m

. This allows us to treat the ratio as a single variable, making it much easier to apply the condition for equal roots.

Why does 'parallel lines' imply 'equal roots' for the quadratic?

Because the lines are parallel, they share the same direction. In 3D geometry, the direction cosines

(l, m, n)

define the direction of a line. If the lines are parallel, their direction cosines are proportional, meaning the ratio

l / m

must be unique. A quadratic equation has a unique solution only when its roots are equal.

What if

m = 0

when we divide by

m^{2}

That is a sharp observation! If

m = 0

, then from

l + m - n = 0

, we get

l = n

. Substituting this into

3 l^{2} + m^{2} + c n l = 0

gives

3 l^{2} + c (l) (l) = 0

, or

l^{2} (3 + c) = 0

. Since

l

cannot be zero for a valid direction cosine, this would imply

c = - 3

. However, the problem asks for the positive value of

c

, and our discriminant method covers the general case where

m

is non-zero.

Animated Solution for Mathematics - Three Dimensional Geometry: If two straight lines whose direction cosines are given by the relations $l + m - n = 0, 3 l^{2} + m^{2} + c n l = 0$ are parallel, then the positive value of $c$ is :

Select Answer:

Visualized Solution

Given Relations

Isolating $n$

Substitution

Expansion

Grouping Terms

Forming a Quadratic in $\frac{l}{m}$

Parallel Lines Condition

Discriminant $D = 0$

Solving for $c$

Final Answer

The Sigma Insight: Direction Cosines and Direction Ratios

Analyzing the Setup

The Art of Reduction

The Transformation

The Final Insight

Similar Questions

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^{2} = m^{2} + n^{2}$ is

The angle between the straight lines, whose direction cosines are given by the equations $2 l + 2 m - n = 0$ and $mn + n l + l m = 0$ , is :

Let $α$ be the angle between the lines whose direction cosines satisfy the equations $l + m - n = 0$ and $l^{2} + m^{2} - n^{2} = 0$ . Then the value of $sin^{4} α + cos^{4} α$ is :

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Select Answer:

Visualized Solution

Given Relations

Isolating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Substitution

Expansion

Grouping Terms

Forming a Quadratic in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ml​

Parallel Lines Condition

Discriminant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } D=0

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c

Final Answer

The Sigma Insight: Direction Cosines and Direction Ratios

Analyzing the Setup

The Art of Reduction

The Transformation

The Final Insight

Similar Questions

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Forming a Quadratic in $\frac{l}{m}$

Discriminant $D = 0$

Solving for $c$

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^{2} = m^{2} + n^{2}$ is

The angle between the straight lines, whose direction cosines are given by the equations $2 l + 2 m - n = 0$ and $mn + n l + l m = 0$ , is :

Let $α$ be the angle between the lines whose direction cosines satisfy the equations $l + m - n = 0$ and $l^{2} + m^{2} - n^{2} = 0$ . Then the value of $sin^{4} α + cos^{4} α$ is :

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