Question

If the length of perpendicular drawn from the point $(4,1)$ on the line $3x - 4y + k = 0$ is $2$ units, then the values of $k$ are

• A. $2,\,-18$
• B. $-2,\,-18$
• C. $-2,\,1$
• D. $-2,\,18$

Hint:

Always remember the distance formula from a point to a line when solving coordinate geometry problems.

Answer 1

The Correct Option is A

Step 1: Using the formula for distance from a point to a line.
The distance of a point $(x_1,y_1)$ from the line $Ax + By + C = 0$ is \[ \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Step 2: Substituting the given values.
Here, $A = 3$, $B = -4$, $C = k$, and the point is $(4,1)$. \[ \frac{|3(4) - 4(1) + k|}{\sqrt{3^2 + (-4)^2}} = 2 \] \[ \frac{|12 - 4 + k|}{5} = 2 \] \[ |8 + k| = 10 \]
Step 3: Solving for $k$.
\[ 8 + k = 10 \Rightarrow k = 2 \] \[ 8 + k = -10 \Rightarrow k = -18 \]
Step 4: Conclusion.
The values of $k$ are $2$ and $-18$.