JEE Main 2011

LEVELJEE Main

Animated Solution for Mathematics - Straight Lines: The lines $L_{1} : y - x = 0$ and $L_{2} : 2 x + y = 0$ intersect the line $L_{3} : y + 2 = 0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$ . Statement-1: The ratio $P R : R Q$ equals $22 : 5$ . Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

Q: Wait, why do we use the Angle Bisector Theorem instead of finding the coordinates of $R$?

You absolutely could find the coordinates of $R$ by finding the equation of the angle bisector and intersecting it with $L_3$, but that is a long, algebraic path prone to calculation errors. The Angle Bisector Theorem is a powerful geometric shortcut that directly gives you the ratio without needing the exact coordinates of $R$. In competitive exams like JEE, recognizing such theorems saves precious time and reduces the chance of silly mistakes.

Select Answer:

Progressive Hints

Hint 1

The Spark: Visualize the geometry of the three lines.

L_{1}

and

L_{2}

intersect at the origin, forming a vertex

A

L_{3}

is a horizontal line

y = - 2

. These three lines create a triangle

△ A P Q

. Your first task is to find the coordinates of vertices

A

P

, and

Q

Hint 2

The Arsenal: Once you have the vertices, you need to find the ratio

P R : R Q

. The most elegant way is to use the Internal Angle Bisector Theorem, which states that the bisector of an angle in a triangle divides the opposite side in the ratio of the lengths of the other two sides:

\frac{P R}{R Q} = \frac{A P}{A Q}

Hint 3

The Execution: Calculate the lengths

A P

and

A Q

using the distance formula from the origin

(0, 0)

to the points

P

and

Q

. Once you have these lengths, the ratio

\frac{P R}{R Q}

is simply

\frac{A P}{A Q}

. Remember to verify if the angle is acute using the dot product of vectors

A P

and

A Q

Visualized Solution

Visualizing the Given Lines

Given lines in the coordinate plane:
$L_{1} : y - x = 0 ⟹ y = x$
$L_{2} : 2 x + y = 0 ⟹ y = - 2 x$
$L_{3} : y + 2 = 0 ⟹ y = - 2$

Finding Intersection Point $A$

Intersection of $L_{1}$ and $L_{2}$ forms vertex $A$ .
Substitute $y = x$ into $y = - 2 x$ :
$x = - 2 x ⟹ 3 x = 0 ⟹ x = 0$
Point $A$ is at the origin $(0, 0)$ .

Finding Vertices $P$ and $Q$

Point $P = L_{1} \cap L_{3}$ :
Substitute $y = - 2$ into $y = x ⟹ x = - 2$ .
So, $P = (- 2, - 2)$
Point $Q = L_{2} \cap L_{3}$ :
Substitute $y = - 2$ into $y = - 2 x ⟹ x = 1$ .
So, $Q = (1, - 2)$

The Triangle $△ A P Q$

The intersections form $△ A P Q$ .
The base $P Q$ lies entirely on $L_{3}$ .
The sides $A P$ and $A Q$ are segments of $L_{1}$ and $L_{2}$ .

Checking the Acute Angle

The problem specifies the acute angle bisector.
Is $∠ P A Q$ acute or obtuse?
Check dot product of vectors $A P$ and $A Q$ :
$A P = (- 2, - 2)$ , $A Q = (1, - 2)$
$A P \cdot A Q = (- 2) (1) + (- 2) (- 2) = 2 > 0$
Since dot product is positive, $∠ P A Q$ is acute.

Calculating Length of $A P$

Use the distance formula for $A (0, 0)$ and $P (- 2, - 2)$ :
$A P = (- 2 - 0)^{2} + (- 2 - 0)^{2}$
$A P = 4 + 4 = 8$
$A P = 22$

Calculating Length of $A Q$

Use the distance formula for $A (0, 0)$ and $Q (1, - 2)$ :
$A Q = (1 - 0)^{2} + (- 2 - 0)^{2}$
$A Q = 1 + 4$
$A Q = 5$

Internal Angle Bisector Theorem

The bisector of $∠ P A Q$ intersects base $P Q$ at $R$ .
Theorem: An internal angle bisector of a triangle divides the opposite side internally in the ratio of the corresponding sides.
Therefore, $\frac{P R}{R Q} = \frac{A P}{A Q}$

Evaluating Statement 1

Substitute the calculated lengths:
$A P = 22$
$A Q = 5$
$\frac{P R}{R Q} = \frac{2 2}{5}$
Thus, Statement-1 is True.

Evaluating Statement 2

Statement-2: "In any triangle, bisector of an angle divides the triangle into two similar triangles."
The bisector creates $△ A P R$ and $△ A QR$ .
These triangles share the same height but have different bases ( $P R \neq = R Q$ ).
They are not similar unless $△ A P Q$ is isosceles ( $A P = A Q$ ).
Thus, Statement-2 is False.

Final Conclusion

Statement-1 is True.
Statement-2 is False.
Correct Option: Statement-1 is true, Statement-2 is false.

00:00 / 00:00

The Sigma Insight: Angle Between Two Lines

Solution Diagram

Analyzing the Setup

The given lines are

L_{1} : y = x

L_{2} : y = - 2 x

, and

L_{3} : y = - 2

. These lines form a triangle

△ A P Q

To find the vertices, we solve for the intersection points: 1. Intersection of

L_{1}

and

L_{2}

x = - 2 x \Rightarrow 3 x = 0 \Rightarrow x = 0

. Thus,

A = (0, 0)

. 2. Intersection of

L_{1}

and

L_{3}

x = - 2

. Thus,

P = (- 2, - 2)

. 3. Intersection of

L_{2}

and

L_{3}

- 2 x = - 2 \Rightarrow x = 1

. Thus,

Q = (1, - 2)

The vertices of the triangle are

A (0, 0)

P (- 2, - 2)

, and

Q (1, - 2)

Verifying the Angle

We must confirm that the angle

∠ P A Q

is acute. We define the vectors:

A P = (- 2, - 2)

A Q = (1, - 2)

The dot product is calculated as:

A P \cdot A Q = (- 2) (1) + (- 2) (- 2) = - 2 + 4 = 2

Since the dot product is positive, the angle

∠ P A Q

is indeed acute.

The Internal Angle Bisector Theorem

The Internal Angle Bisector Theorem states that the bisector of

∠ P A Q

divides the opposite side

P Q

at a point

R

such that the ratio of the segments is equal to the ratio of the adjacent sides:

\frac{P R}{R Q} = \frac{A P}{A Q}

We calculate the lengths of the sides

A P

and

A Q

A P = (- 2 - 0)^{2} + (- 2 - 0)^{2} = 4 + 4 = 8 = 22

A Q = (1 - 0)^{2} + (- 2 - 0)^{2} = 1 + 4 = 5

Substituting these values into the ratio, we obtain:

\frac{P R}{R Q} = \frac{2 2}{5}

This confirms that the geometric property described in Statement-1 is true.

Evaluating Statement-2

Statement-2 claims that an angle bisector always divides a triangle into two similar triangles.

Consider the triangles

△ A P R

and

△ A QR

formed by the bisector. While they share the same altitude from

A

to the line

P Q

, they are not necessarily similar.

Similarity requires the triangles to have equal corresponding angles. This condition is only satisfied if the original triangle

△ A P Q

is isosceles (where

A P = A Q

). Since this is not true for all triangles, Statement-2 is false.

Common Traps & FAQs

Wait, why do we use the Angle Bisector Theorem instead of finding the coordinates of

R

You absolutely could find the coordinates of

R

by finding the equation of the angle bisector and intersecting it with

L_{3}

, but that is a long, algebraic path prone to calculation errors. The Angle Bisector Theorem is a powerful geometric shortcut that directly gives you the ratio without needing the exact coordinates of

R

. In competitive exams like JEE, recognizing such theorems saves precious time and reduces the chance of silly mistakes.

How do I know if the angle is acute or obtuse without drawing it?

That is a fantastic question! You don't need to rely on a perfect sketch. Use the dot product of the vectors forming the angle. If

A P \cdot A Q > 0

, the angle is acute. If it is negative, the angle is obtuse. In this problem, the dot product is positive, confirming the angle is acute, so we are using the correct bisector.

Is Statement 2 always false?

Yes, in the context of general triangles. An angle bisector divides the opposite side into segments proportional to the adjacent sides, but it does not necessarily create similar triangles. The two resulting triangles,

△ A P R

and

△ A QR

, share the same height from

A

, but their bases

P R

and

R Q

are generally unequal. They would only be similar if the original triangle were isosceles, where the bisector is also the altitude. Therefore, as a general statement for 'any triangle', it is false.

Select Answer:

Visualized Solution

Visualizing the Given Lines

Finding Intersection Point $A$

Finding Vertices $P$ and $Q$

The Triangle $△ A P Q$

Checking the Acute Angle

Calculating Length of $A P$

Calculating Length of $A Q$

Internal Angle Bisector Theorem

Evaluating Statement 1

Evaluating Statement 2

Final Conclusion

The Sigma Insight: Angle Between Two Lines

Analyzing the Setup

Verifying the Angle

The Internal Angle Bisector Theorem

Evaluating Statement-2

Similar Questions

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Let $P QR$ be a right angled isosceles triangle, right angled at $P (2, 1)$ . If the equation of the line $QR$ is $2 x + y = 3$ , then the equation representing the pair of lines $P Q$ and $P R$ is

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Select Answer:

Visualized Solution

Visualizing the Given Lines

Finding Intersection Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Finding Vertices .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Q

The Triangle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } △APQ

Checking the Acute Angle

Calculating Length of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AP

Calculating Length of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AQ

Internal Angle Bisector Theorem

Evaluating Statement 1

Evaluating Statement 2

Final Conclusion

The Sigma Insight: Angle Between Two Lines

Analyzing the Setup

Verifying the Angle

The Internal Angle Bisector Theorem

Evaluating Statement-2

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A(1,0),B(2,−1) and C(37​,34​) be three points. If the equation of the bisector of the angle ABC is αx+βy=5, then the value of α2+β2 is

Finding Intersection Point $A$

Finding Vertices $P$ and $Q$

The Triangle $△ A P Q$

Calculating Length of $A P$

Calculating Length of $A Q$

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Let $P QR$ be a right angled isosceles triangle, right angled at $P (2, 1)$ . If the equation of the line $QR$ is $2 x + y = 3$ , then the equation representing the pair of lines $P Q$ and $P R$ is

If the line $l_{1} : 3 y - 2 x = 3$ is the angular bisector of the lines $l_{2} : x - y + 1 = 0$ and $l_{3} : α x + β y + 17 = 0$ , then $α^{2} + β^{2} - α - β$ is equal to

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