Question

The number of lines passing through A (3, 4) and the sum of whose non-zero intercepts is zero is (are)

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(4\)

Hint:

Write the intercept form with intercepts a and -a and use the point.

Answer 1

The Correct Option is A

Step 1: Parametrise by slope:
A line through (3,4): $y - 4 = m(x - 3)$. The x-intercept is $3 - 4/m$, the y-intercept is $4 - 3m$.

Step 2: Sum zero:
$3 - \dfrac4m + 4 - 3m = 0 \Rightarrow 7m - 4 - 3m^2 = 0 \Rightarrow 3m^2 - 7m + 4 = 0$, so $m = 1$ or $m = \tfrac43$.

Step 3: Filter:
$m = 4/3$ gives x-intercept $0$ and y-intercept $0$, a line through the origin with no nonzero intercepts. Only $m = 1$ is valid, so there is 1 line.

Final Answer:
Only one line qualifies. \[ \boxed{\text{(A) }1} \]