Question

The graphs of ax+by = c, dx+ey = f will be,
(A) Parallel if the system has no solution
(B) Coincident if the system has finite number of solutions
(C) Intersecting if the system has two solutions
(D) Intersecting if the system has only one solution

• A. A and B only
• B. B and C only
• C. A and D only
• D. B and D only

Answer 1

The Correct Option is C

To solve this problem, we need to analyze the relationship between two linear equations in two variables, which are given as:

• \(ax + by = c\)
• \(dx + ey = f\)

We are interested in finding the relationship between these two lines in terms of their interaction according to the possible scenarios:

1. Two lines are parallel if they have no point of intersection. This happens when their slopes are equal but their y-intercepts are different. The condition for this is: \(\frac{a}{d} = \frac{b}{e} \neq \frac{c}{f}\). Thus, option A (Parallel if the system has no solution) corresponds to this and is correct.
2. Two lines are coincident if every point on one line is a point on the other line. This implies they have an infinite number of solutions. The condition for this is: \(\frac{a}{d} = \frac{b}{e} = \frac{c}{f}\). This scenario is not stated in any of the options.
3. Two lines intersect at exactly one point if they have different slopes. This means the system has a unique solution, and the condition is: \(\frac{a}{d} \neq \frac{b}{e}\). Option D (Intersecting if the system has only one solution) correctly describes this scenario.

Now, let's eliminate the incorrect options:

• Option B mentions intersecting if the system has two solutions, which is incorrect as linear equations in two variables cannot have exactly two solutions.
• Therefore, Option "A and D only" is the correct choice, as it correctly identifies the conditions for parallel lines with no solution and intersecting lines with a unique solution.

Answer 2

The equations \( ax + by = c \) and \( dx + ey = f \) represent straight lines in the Cartesian plane. The nature of the relationship between these two lines depends on the values of the coefficients.

1. If the lines are parallel, they will never intersect. This happens if the system has no solution. This corresponds to the case where the slopes of the lines are equal. For this to happen, the ratios of the coefficients must satisfy: \[ \frac{a}{d} = \frac{b}{e} \] In this case, the lines are parallel and do not intersect.
2. If the lines are coincident, they will coincide at every point, and the system will have infinite solutions. This occurs if the two equations represent the same line, which can happen if the ratios of the coefficients are the same and the constants are also proportional: \[ \frac{a}{d} = \frac{b}{e} = \frac{c}{f} \] In this case, the lines coincide.
3. If the lines are intersecting, they will meet at exactly one point, and the system will have one solution. This happens if the ratios of the coefficients are not equal, and the lines have different slopes. Thus, the lines will intersect at exactly one point.

Given that Statement A states that the lines are parallel when there is no solution, and Statement D states that the lines intersect at exactly one solution, the correct answer is (3), which corresponds to the case where the lines are either parallel (no solution) or intersecting (one solution).