33. Assertion \( (A) \) : Every line represented by linear equation in two variables of type ax + by \( +\mathrm{c}=0(\mathrm{a} \neq 0, \mathrm{~b} \neq 0, \mathrm{c} \neq 0) \) always meet both the axes ( \( x \) - axis and \( y \) - axis) exactly at one point. Reason (R): Line represented by equation ax + by \( +\mathrm{c}=0 \) cuts \( x \) - axis at \( -\frac{c}{a} \) and \( y \) - axis at \( -\frac{c}{b} \) as equation of \( x \) - axis is \( y=0 \) and equation \( y \) - axis is \( x=0 \)

To analyze the assertion (A) and the reason (R), let's break them down: **Assertion (A)**: Every line represented by a linear equation in two variables of the form \( ax + by + c = 0 \) (where \( a \neq 0, b \neq 0, c \neq 0 \)) always meets both the axes (the \( x \)-axis and the \( y \)-axis) exactly at one point. **Reason (R)**: The line represented by the equation \( ax + by + c = 0 \) cuts the \( x \)-axis at \( -\frac{c}{a} \) and the \( y \)-axis at \( -\frac{c}{b} \), since the equation of the \( x \)-axis is \( y = 0 \) and the equation of the \( y \)-axis is \( x = 0 \). ### Analysis: 1. **Finding the intersection with the axes**: - To find the intersection with the \( x \)-axis, set \( y = 0 \): \[ ax + c = 0 \implies x = -\frac{c}{a} \] - To find the intersection with the \( y \)-axis, set \( x = 0 \): \[ by + c = 0 \implies y = -\frac{c}{b} \] 2. **Conditions for intersections**: - Since \( a \neq 0 \) and \( c \neq 0 \), \( -\frac{c}{a} \) is a valid point on the \( x \)-axis. - Since \( b \neq 0 \) and \( c \neq 0 \), \( -\frac{c}{b} \) is a valid point on the \( y \)-axis. - Therefore, the line will intersect both axes at distinct points. 3. **Conclusion**: - The assertion (A) is true because the line will indeed intersect both axes at exactly one point each, given the conditions \( a \neq 0 \), \( b \neq 0 \), and \( c \neq 0 \). - The reason (R) correctly explains how the intersections are found and confirms the assertion. ### Final Answer: Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion.