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: Understanding the Concept:
If the nonzero intercepts on the axes are \(a\) and \(b\), the line is \(\dfrac{x}{a} + \dfrac{y}{b} = 1\). The sum is zero when \(b = -a\).

Step 2: Form the equation:
\[ \frac{x}{a} + \frac{y}{-a} = 1 \Rightarrow x - y = a \]

Step 3: Use the point (3, 4):
\[ 3 - 4 = a \Rightarrow a = -1 \]
The line is \(x - y = -1\), with intercepts \(-1\) and \(1\), which sum to zero.
The line through (3, 4) and the origin has both intercepts equal to zero, so it has no nonzero intercepts and does not count here.

Step 4: Final count:
Only one such line exists.

Final Answer:
Exactly one line, x - y = -1, fits. \[ \boxed{\text{(A) }1} \]