Problem 79 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A straight line through the point \(A(3,4)\) is such that its intercept between the axes is bisected at \(A\). Its equation is (A) \(x+y=7\) (B) \(3 x-4 y+7=0\) (C) \(4 x+3 y=24\) (D) \(3 x+4 y=25\)

Step-by-Step Solution

Answer

The equation of the line is \(4x + 3y = 24\), which corresponds to option (C).

Step 1: Understanding the Line's Characteristics

Since the intercept between the axes is bisected at \(A(3,4)\), the line's equation can be expressed in intercept form: \(\frac{x}{a} + \frac{y}{b} = 1\). The coordinates \((a, 0)\) and \((0, b)\) are the intercepts and the midpoint of these intercepts is \(A(3,4)\).

Step 2: Relate Intercepts and Midpoint

The coordinates of the midpoint of the line segment with intercepts \((a,0)\) and \((0,b)\) are \(\left(\frac{a+0}{2}, \frac{0+b}{2}\right) = \left(\frac{a}{2}, \frac{b}{2}\right)\). Set these equal to \((3, 4)\), yielding \(\frac{a}{2} = 3\) and \(\frac{b}{2} = 4\).

Step 3: Solve for Intercept Values

From \(\frac{a}{2} = 3\), solve to get \(a = 6\). From \(\frac{b}{2} = 4\), solve to get \(b = 8\).

Step 4: Write the Equation of the Line

Substitute back into the intercept form: \(\frac{x}{6} + \frac{y}{8} = 1\). To convert this into standard linear form, clear the fractions by multiplying through by the common denominator, 24. This gives: \(4x + 3y = 24\).

Step 5: Match with Given Options

The derived equation \(4x + 3y = 24\) matches option (C).

Key Concepts

Midpoint of a Line Segment

The midpoint of a line segment is a fundamental concept in coordinate geometry. It is the point that divides the segment into two equal halves. Let's imagine a line segment that runs between two points \( (x_1, y_1) \) and \( (x_2, y_2) \). To find the midpoint \( M \) of this segment, we use the formula:

• \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

Essentially, you calculate the average of the x-coordinates and the average of the y-coordinates.
For instance, if the endpoints are at \( (1, 5) \) and \( (7, 9) \), the midpoint will be \( M = \left( \frac{1 + 7}{2}, \frac{5 + 9}{2} \right) = (4, 7) \).
In the exercise we discussed, knowing the midpoint \( A(3,4) \) between intercepts \( (a, 0) \) and \( (0, b) \) allowed us to determine the intercept values themselves, paving the way for finding the line equation.

Equation of a Line

Finding the equation of a line helps in understanding its path and behavior on a coordinate plane. Typically, lines are expressed in several forms, like slope-intercept and intercept forms. In this particular case, we're focusing on the 'intercept form' which is expressed as:

• \( \frac{x}{a} + \frac{y}{b} = 1 \)

Here, \( a \) and \( b \) represent the intercepts on the x-axis and y-axis, respectively. This form is especially useful when you know where a line crosses the axes but not necessarily the slope or precise point details.
For our example in the exercise, once we computed the intercepts \( a = 6 \) and \( b = 8 \) from the midpoint condition, the intercept form proved invaluable. By substituting these intercepts into the form, we obtained:

• \( \frac{x}{6} + \frac{y}{8} = 1 \)

Converting it to the standard form resulted in a clearer equation among the given options. Remember, understanding different forms of line equations gives you the flexibility to solve a variety of geometric problems efficiently.

Coordinate Geometry

Coordinate geometry, or analytic geometry, merges algebra with geometry, using coordinates to solve geometric problems. This approach uses a coordinate plane, which consists of two axes, the x-axis and y-axis, intersecting at the origin.Any point on this plane can be described by a pair of numbers \( (x, y) \), representing horizontal and vertical distances from the origin. This system enables precise representation of geometric shapes, lines, and curves.

• Applications: Commonly used for solving problems related to distances, midpoints, and slopes.
• Advantages: Provides a clear visual understanding and allows algebraic equations to define geometric figures.

In reference to our exercise, coordinate geometry allowed us to find and set up the relations between intercepts of a line and a geometric property, the midpoint. These connections demonstrate how algebraic equations can represent geometric properties in detail, solving questions related to locations and paths of geometric figures.