Graphical Representation Using Ratio of Coefficients

Video Explanation

Question

Gloria is walking along the path joining the points \((-2, 3)\) and \((2, -2)\), while Suresh is walking along the path joining \((0, 5)\) and \((4, 0)\). Represent this situation graphically and check the nature of the paths using the ratio of coefficients.

Solution

Step 1: Find the Equation of Gloria’s Path

Slope of the line:

\[ m_1 = \frac{-2 – 3}{2 – (-2)} = \frac{-5}{4} \]

Equation of the line:

\[ y – 3 = -\frac{5}{4}(x + 2) \]

\[ 4y – 12 = -5x – 10 \]

\[ 5x + 4y – 2 = 0 \quad \text{(Equation 1)} \]

Step 2: Find the Equation of Suresh’s Path

Slope of the line:

\[ m_2 = \frac{0 – 5}{4 – 0} = \frac{-5}{4} \]

Equation of the line:

\[ y – 5 = -\frac{5}{4}(x – 0) \]

\[ 4y – 20 = -5x \]

\[ 5x + 4y – 20 = 0 \quad \text{(Equation 2)} \]

Step 3: Compare the Ratios of Coefficients

Equation (1): \(5x + 4y – 2 = 0\)

Equation (2): \(5x + 4y – 20 = 0\)

Now,

\[ \frac{a_1}{a_2} = \frac{5}{5} = 1, \quad \frac{b_1}{b_2} = \frac{4}{4} = 1, \quad \frac{c_1}{c_2} = \frac{-2}{-20} = \frac{1}{10} \]

Step 4: Interpretation

Since

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \quad \text{but} \quad \frac{a_1}{a_2} \neq \frac{c_1}{c_2}, \]

the two straight lines are parallel.

Conclusion

The paths of Gloria and Suresh are represented by two parallel straight lines.

Hence, they are walking along parallel paths and will never meet.