Problem 74 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\theta_{1}\) be the angle between two lines \(2 x+3 y+c_{1}=0\) and \(-x+5 y+c_{2}=0\) and \(\theta_{2}\) be the angle between two lines \(2 x+3 y+c_{1}=0\) and \(-x+5 y+c_{3}=0\), where \(c_{1}, c_{2}, c_{3}\) are any real numbers: Statement-1: If \(c_{2}\) and \(c_{3}\) are proportional, then \(\theta_{1}=\theta_{2}\). Statement-2: \(\theta_{1}=\theta_{2}\) for all \(c_{2}\) and \(c_{3}\). \mathrm{\\{} O n l i n e ~ A p r i l ~ 2 3 , ~ 2 0 1 3 ] ~ (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation of Statement-1. (b) Statement= 1 is true, Statement 2 is true; Statement- 2 is not a correct explanation of Statement-1. (c) Statement- 1 is false; Statement- 2 is true. (d) Statement- 1 is true; Statement- 2 is false.

Step-by-Step Solution

Answer

(b) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement-1.

Step 1: Understand the Relationship Between Angles of Lines

The angle \( \theta \) between two lines \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) is given by the formula \( \tan \theta = \left| \frac{a_1b_2 - a_2b_1}{a_1a_2 + b_1b_2} \right| \). In our problem, the lines' equation coefficients for calculating \( \theta_1 \) and \( \theta_2 \) do not include \( c_1, c_2, \) or \( c_3 \).

Step 2: Evaluate Expression for θ₁

Calculate \( \tan \theta_1 \) for the lines \( 2x + 3y + c_1 = 0 \) and \( -x + 5y + c_2 = 0 \). Using the formula from Step 1: \[ \tan \theta_1 = \left| \frac{2 \cdot 5 - (-1) \cdot 3}{2 \cdot (-1) + 3 \cdot 5} \right| = \left| \frac{10 + 3}{-2 + 15} \right| = \left| \frac{13}{13} \right| = 1. \]

Step 3: Evaluate Expression for θ₂

Calculate \( \tan \theta_2 \) for the lines \( 2x + 3y + c_1 = 0 \) and \( -x + 5y + c_3 = 0 \). Using the same method as \( \theta_1 \), we obtain: \[ \tan \theta_2 = \left| \frac{2 \cdot 5 - (-1) \cdot 3}{2 \cdot (-1) + 3 \cdot 5} \right| = \left| \frac{13}{13} \right| = 1. \]

Step 4: Analyze the Statements

Both \( \theta_1 = \theta_2 \) are equal since their expressions evaluate the same regardless of \( c_1, c_2, \) or \( c_3 \). Thus, Statement-1 is true: \( \theta_1 = \theta_2 \) if \( c_2 \) and \( c_3 \) are proportional. Statement-2 is also true, because \( \theta_1 = \theta_2 \) for all such \( c_2 \) and \( c_3 \). However, Statement-2 is not a valid explanation for Statement-1.

Step 5: Conclusion

Based on the evaluation above, although both statements are true, Statement-2 is not a logical explanation for Statement-1. Therefore, the correct choice is (b).

Key Concepts

Angle Between Lines

To find the angle between two lines in analytic geometry, we use the angle \( \theta \) between the lines' slopes. This angle is crucial since it helps us understand how two lines are oriented in relation to one another. We calculate it using the formula:
\( \tan \theta = \left| \frac{a_1b_2 - a_2b_1}{a_1a_2 + b_1b_2} \right| \).
- Here, the coefficients \( a_1, b_1 \) and \( a_2, b_2 \) come from the equations of the respective lines \( a_1x + b_1y + c = 0 \) and \( a_2x + b_2y + c = 0 \).
- Note that the constant terms \( c_1, c_2 \) and \( c_3 \) do not affect the calculation of \( \theta \). This is because these constants shift the line vertically or horizontally but do not affect their slope.
When the tangents are equal, \( \tan \theta_1 = \tan \theta_2 = 1 \), indicating the angles are the same. This suggests that the lines have the same inclination relative to each other.

Equation of a Line

The equation of a line in two-dimensional space is usually expressed in the form \( ax + by + c = 0 \). Each component has a special role:
- The coefficients \( a \) and \( b \) determine the line's slope, which is crucial for understanding the direction of the line. The slope \( m \) can be found by rearranging the equation to the form \( y = mx + c \), where \( m = -\frac{a}{b} \).- The constant \( c \) represents the line's intersection with the axes. It affects the position of the line without changing its direction.
By comparing the coefficients of two line equations, we can predict how they relate to one another. If we multiply the coefficients of one equation by the same factor to get the other equation, these lines are proportional, meaning \( \theta \) remains unchanged between them.

Tangent of an Angle

The tangent of an angle is a fundamental concept in trigonometry. It represents the ratio of the opposite side to the adjacent side in a right triangle. For two intersecting lines, the tangent helps determine the steepness of one line relative to the other.
- When considering angles between lines, the tangent formula for angles emphasizes the differences in their slopes:
\( \tan \theta = \left| \frac{a_1b_2 - a_2b_1}{a_1a_2 + b_1b_2} \right| \)
This formula derives from combining the lines' slopes and utilizes the tangent to offer a clearer depiction of how sharply they intersect. A calculated tangent of \( \theta \), like 1, shows that the lines create a 45-degree angle with each other, demonstrating symmetry in intersection.