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AI-তৈরি নমুনা উত্তর

1. Point of Intersection:
   The given lines are:
   $2x - y = 1 \implies y = 2x - 1$ ...(i)
   $3x - 4y + 6 = 0$ ...(ii)

   Substituting (i) into (ii):
   $3x - 4(2x - 1) + 6 = 0$
   $3x - 8x + 4 + 6 = 0$
   $-5x + 10 = 0 \implies x = 2$

   From (i), $y = 2(2) - 1 = 3$.
   So, the point of intersection is $(2, 3)$.

2. Equation of the Parallel Line:
   The required line is parallel to $4x + 3y - 6 = 0$.
   Therefore, its equation is of the form:
   $4x + 3y + k = 0$

   Since this line passes through the point $(2, 3)$:
   $4(2) + 3(3) + k = 0$
   $8 + 9 + k = 0 \implies k = -17$

   Hence, the equation of the line is:
   $4x + 3y = 17$

3. Finding $a$ and $b$:
   Dividing both sides by 17 to put it into the intercept form $\frac{x}{a} + \frac{y}{b} = 1$:
   $\frac{4x}{17} + \frac{3y}{17} = 1$
   $\implies \frac{x}{\frac{17}{4}} + \frac{y}{\frac{17}{3}} = 1$

   Comparing with $\frac{x}{a} + \frac{y}{b} = 1$, we get:
   $a = \frac{17}{4}$
   $b = \frac{17}{3}$