The number of lines passing through A(3, 4) s | vedclass

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(A) Let the equation of the line be $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are the $x$ and $y$ intercepts respectively.Given that the sum

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(A) Let the equation of the line be $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are the $x$ and $y$ intercepts respectively.Given that the sum of intercepts is zero, $a + b = 0$, which implies $b = -a$.Substituting this into the line equation: $\frac{x}{a} - \frac{y}{a} = 1$, or $x - y = a$.Since the line passes through $A(3, 4)$, we substitute these coordinates: $3 - 4 = a$, so $a = -1$.This gives the line $x - y = -1$, or $x - y + 1 = 0$.Also, consider the case where the line passes through the origin $(0, 0)$. If the line passes through $(0, 0)$ and $(3, 4)$, its equation is $y = \frac{4}{3}x$, or $4x - 3y = 0$. Here, the intercepts are $a=0$ and $b=0$. However, the problem specifies non-zero intercepts.Thus, only one such line exists: $x - y + 1 = 0$.

The number of lines passing through $A(3, 4)$ such that the sum of their non-zero intercepts is zero is:

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