Question:
If the straight lines $x + 2y - 9 = 0$, $3x + 5y - 5 = 0$ and $ax + by - 1 = 0$ are concurrent, then the straight line $22x - 35y = 1$ passes through the point:
If the straight lines $x + 2y - 9 = 0$, $3x + 5y - 5 = 0$ and $ax + by - 1 = 0$ are concurrent, then the straight line $22x - 35y = 1$ passes through the point:
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Look at the coefficients of the given line $22x - 35y = 1$ and the obtained relation $22b - 35a = 1$. By matching variables, $x = b$ and $y = a$, yielding the point $(b, a)$ instantly.
Updated On: Jun 3, 2026
- $(a, b)$
- $(b, a)$
- $(-a, -b)$
- $(-b, -a)$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Three straight lines are concurrent if they intersect at a single, common point.
Step 2: Meaning
We solve the first two line equations to find their point of intersection, and then substitute this point into the third line equation since they are concurrent.
Step 3: Analysis
The first two equations are: 1) $x + 2y - 9 = 0 \implies x = 9 - 2y$ 2) $3x + 5y - 5 = 0$ Substituting (1) into (2): \[ 3(9 - 2y) + 5y - 5 = 0 \implies 27 - 6y + 5y - 5 = 0 \implies 22 - y = 0 \implies y = 22 \] Substituting $y = 22$ back into (1): \[ x = 9 - 2(22) = 9 - 44 = -35 \] Thus, the point of intersection is $(-35, 22)$. Since the third line $ax + by - 1 = 0$ passes through this point: \[ a(-35) + b(22) - 1 = 0 \implies 22b - 35a = 1 \] Now, we are given the equation of a straight line $22x - 35y = 1$. Substituting the point $(b, a)$ into this equation: \[ 22(b) - 35(a) = 1 \implies 22b - 35a = 1 \] This perfectly satisfies the condition.
Step 4: Conclusion
Therefore, the line $22x - 35y = 1$ passes through the point $(b, a)$, corresponding to option (B).
Final Answer: (B)
Three straight lines are concurrent if they intersect at a single, common point.
Step 2: Meaning
We solve the first two line equations to find their point of intersection, and then substitute this point into the third line equation since they are concurrent.
Step 3: Analysis
The first two equations are: 1) $x + 2y - 9 = 0 \implies x = 9 - 2y$ 2) $3x + 5y - 5 = 0$ Substituting (1) into (2): \[ 3(9 - 2y) + 5y - 5 = 0 \implies 27 - 6y + 5y - 5 = 0 \implies 22 - y = 0 \implies y = 22 \] Substituting $y = 22$ back into (1): \[ x = 9 - 2(22) = 9 - 44 = -35 \] Thus, the point of intersection is $(-35, 22)$. Since the third line $ax + by - 1 = 0$ passes through this point: \[ a(-35) + b(22) - 1 = 0 \implies 22b - 35a = 1 \] Now, we are given the equation of a straight line $22x - 35y = 1$. Substituting the point $(b, a)$ into this equation: \[ 22(b) - 35(a) = 1 \implies 22b - 35a = 1 \] This perfectly satisfies the condition.
Step 4: Conclusion
Therefore, the line $22x - 35y = 1$ passes through the point $(b, a)$, corresponding to option (B).
Final Answer: (B)
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