11. Consider the graphs of the two linear equations ax + by = c and bx – ay = c, where a, b
and c are all greater than zero. These graphs
a. are perpendicular
b. are parallel
c. intersect at an acute angle
d. intersect at more than one point
The graphs of the linear equations ax + by = c and bx – ay = c, where a, b, c > 0, represent straight lines that intersect at a single point. Converting to slope-intercept form reveals their slopes as m₁ = -a/b and m₂ = b/a, satisfying m₁ × m₂ = -1, confirming perpendicularity. They intersect at (c(a + b)/(a² + b²), c(b – a)/(a² + b²)) [execute_python].
Option Analysis
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a. Are perpendicular: Correct. Slopes multiply to -1 (-a/b × b/a = -1), defining perpendicular lines since a, b > 0 ensures defined, non-zero slopes.
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b. Are parallel: Incorrect. Parallel lines share identical slopes, but m₁ ≠ m₂ unless a = b (still perpendicular as product = -1); generally distinct.
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c. Intersect at an acute angle: Incorrect. Perpendicular lines form 90° (right angle), not acute (<90°); slope condition proves exact orthogonality.
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d. Intersect at more than one point: Incorrect. Distinct non-parallel lines intersect exactly once, as solving yields unique solution; same c shifts but doesn’t coincide [execute_python].
Key Phrase Integration
In graphs of ax + by = c and bx – ay = c perpendicular scenarios, CSIR NET aspirants note the rotated coefficient pattern (a↔b, b→-a) creates orthogonal lines. This tests slope product rule: for lines y = m₁x + c₁ and y = m₂x + c₂, perpendicular if m₁m₂ = -1. Practice verifies via substitution or determinants.
Exam Preparation Tips
Visualize: First line tilts based on -a/b; second, its “90° rotate” via b/a. For variants (c ≠ same), adjust intercepts but perpendicularity holds. Solve systems algebraically: add equations for y, subtract for x, denominator a² + b² > 0 [execute_python]. Master for competitive exams like CSIR NET Life Sciences quantitative sections.
