Question

Distance of the point \( (-2,3) \) from the line \( 12x - 5y - 2 = 0 \) is \( \frac{41}{k} \). Then the value of \( k \) is

• A. \( \sqrt{134} \)
• B. \( \sqrt{13} \)
• C. 13
• D. 1

Hint:

Always use the perpendicular distance formula for point-to-line problems and simplify carefully.

Answer 1

The Correct Option is C

Step 1: Recall the distance formula.
Distance of point \( (x_1,y_1) \) from line \( ax+by+c=0 \) is:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}. \]

Step 2: Identify values.
Given:
\[ a=12,\quad b=-5,\quad c=-2,\quad (x_1,y_1)=(-2,3). \]

Step 3: Substitute into formula.
\[ d = \frac{|12(-2) -5(3) -2|}{\sqrt{12^2 + (-5)^2}}. \]
\[ = \frac{|-24 -15 -2|}{\sqrt{144 + 25}}. \]
\[ = \frac{|-41|}{\sqrt{169}}. \]

Step 4: Simplify numerator and denominator.
\[ d = \frac{41}{13}. \]

Step 5: Compare with given expression.
Given:
\[ d = \frac{41}{k}. \]

Step 6: Equate both expressions.
\[ \frac{41}{k} = \frac{41}{13}. \]
\[ k = 13. \]

Step 7: Final conclusion.
Thus, the value of \( k \) is 13.
Final Answer:
\[ \boxed{13}. \]