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

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).