Question

A straight line makes equal negative intercepts on the coordinate axes. If the perpendicular distance from the origin to the line is 4 units, then the equation of the line is ......

• A. \(x+y+4 = 0\)
• B. \(x+y+4\sqrt{2} = 0\)
• C. \(x-y+4 = 0\)
• D. \(x-y+4\sqrt{2} = 0\)

Hint:

Equal intercepts give slope -1; use the distance-from-origin formula to find the constant.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
A line with equal intercepts \(a\) on both axes has the form \(\dfrac{x}{a}+\dfrac{y}{a}=1\), that is \(x+y=a\). Negative intercepts mean \(a<0\).

Step 2: Write the line
\[ x+y-a=0,\quad a<0 \]
Let \(a=-k\) with \(k>0\). The line is \(x+y+k=0\).

Step 3: Distance from origin
\[ \frac{|k|}{\sqrt{1^2+1^2}}=\frac{k}{\sqrt2}=4\Rightarrow k=4\sqrt2 \]

Step 4: Equation
\[ x+y+4\sqrt2=0 \]

Step 5: Check the options
Option (A) \(x+y+4=0\) is at distance \(4/\sqrt2=2\sqrt2\), not 4. Options (C) and (D) have unequal-sign intercepts, so they do not match. The answer is (B).

Final Answer:
The line is \(x+y+4\sqrt{2}=0\), option (B). \[ \boxed{x+y+4\sqrt{2}=0} \]