Problem 52 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation of the straight line passing through the point \((4,5)\) and making equal angles with the two straight lines given by the equations \(3 x-4 y-7=0\) and \(12 x-5 y+6=0\), is (A) \(9 x-7 y-1=0\) (B) \(9 x+7 y-1=0\) (C) \(7 x+9 y-73=0\) (D) \(7 x+9 y+73=0\)

Step-by-Step Solution

Answer

The equation is (A) \(9x - 7y - 1 = 0\).

Step 1: Understand Given Conditions

We need to find the equation of a line that passes through the point \((4, 5)\) and makes equal angles with two given lines: \(3x - 4y - 7 = 0\) and \(12x - 5y + 6 = 0\).

Step 2: Find Slopes of Given Lines

Calculate the slope of the line \(3x - 4y - 7 = 0\). Rewriting in slope-intercept form, we get \(y = \frac{3}{4}x - \frac{7}{4}\), so the slope is \(\frac{3}{4}\).Calculate the slope of the line \(12x - 5y + 6 = 0\). Rewriting, we have \(y = \frac{12}{5}x + \frac{6}{5}\), so the slope is \(\frac{12}{5}\).

Step 3: Condition for Equal Angles

If a line makes equal angles with two other lines, its slope \(m\) has a relation with the slopes \(m_1\) and \(m_2\) of the given lines: \(\frac{m - m_1}{1 + m m_1} = \pm \frac{m_2 - m}{1 + m m_2}\).

Step 4: Solve the Equation for the Slope of Required Line

Let \(m = \frac{9}{7}\), the slopes \(m_1 = \frac{3}{4}\), and \(m_2 = \frac{12}{5}\).For equal angles, solve \(\frac{\frac{9}{7} - \frac{3}{4}}{1 + \frac{9}{7} \times \frac{3}{4}} = \frac{\frac{12}{5} - \frac{9}{7}}{1 + \frac{12}{5} \times \frac{9}{7}}\).

Step 5: Verify Possible Slopes

Verify which slope \(m = \frac{9}{7}\) results in an equation passing through the point \((4, 5)\). Substitute \((x, y) = (4, 5)\) into each equation to confirm.

Step 6: Check Options Against Conditions

After confirming possible slope, determine which option satisfies the line equation. Plugging in, option \((A) \; 9x - 7y - 1 = 0\) fits \((4, 5)\) when using verified slope \(m = \frac{9}{7}\).

Key Concepts

Slopes

In coordinate geometry, the slope of a line measures its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between any two points on the line. Expressed as the formula, the slope \(m\) can be defined for a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) as:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

For the given exercise, the slopes of the lines \(3x - 4y - 7 = 0\) and \(12x - 5y + 6 = 0\) have been calculated. By rearranging these equations into the form \(y = mx + c\), it's found that their slopes are \(\frac{3}{4}\) and \(\frac{12}{5}\) respectively.
The slope is crucial for understanding the line's behavior in relation to other lines. A smaller slope indicates a gentler incline compared to a larger slope, which represents a steeper line. Slopes can also indicate the direction of a line: positive slopes rise from left to right, while negative slopes fall.

Angles Between Lines

Understanding the angle between two lines is essential when studying coordinate geometry. When two lines are given, the angle \(\theta\) between them can be found through their slopes. If the slopes of the lines are \(m_1\) and \(m_2\), the tangent of the angle between the lines is given by:

• \( \tan\theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \)

In our problem, a unique case is considered: the line needed should make equal angles with two other lines. This unique condition means solving an equation that relates the slopes \(m\), \(m_1\), and \(m_2\). The formula used is:

• \( \frac{m - m_1}{1 + mm_1} = \pm \frac{m_2 - m}{1 + mm_2} \)

This equation essentially states that the chosen slope \(m\) either equals the angle bisector or acts as the supplemental angle to the slope of the angle bisector between the lines. Calculating this successfully determines the direction of the new line.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves using algebraic techniques to solve geometric problems. It is the study of geometry using a coordinate system, allowing algebraic representation of geometrical points and lines.
To find a line's equation passing through a specific point \((x_0, y_0)\) with a known slope \(m\), the point-slope form is useful:

• \( y - y_0 = m(x - x_0) \)

Using these principles, we can find the equation of a line by substituting the required slope \(m = \frac{9}{7}\) and the point \((4, 5)\). Simplifying the point-slope equation results in the line's equation which fits the problem's conditions.
The main power of coordinate geometry is its ability to visually and algebraically handle the complexities arising from geometric configurations. Working through the problem will improve understanding of how algebraic manipulations reveal geometric truths, such as finding points of intersection or determining line paralells.