Question

P is a point on \( x + y + 5 = 0 \), whose perpendicular distance from \( 2x + 3y + 3 = 0 \) is \( \sqrt{13} \), then the coordinates of P are:

• A. \( (20, -25) \)
• B. \( (1, -6) \)
• C. \( (-6,1) \)
• D. \( (\sqrt{13}, -5 - \sqrt{13}) \)

Hint:

- The perpendicular distance formula is useful for solving constraints on points lying on lines. - Always solve for \( y \) in terms of \( x \) when given a line equation to simplify calculations.

Answer 1

The Correct Option is B

Step 1: Equation of the line containing point \( P \). We are given the equation \( x + y + 5 = 0 \), which simplifies to \( y = -x - 5 \). Step 2: Perpendicular distance formula. The formula for the perpendicular distance from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is: \[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}. \] Substituting the values, we get: \[ \frac{|2x_1 + 3y_1 + 3|}{\sqrt{2^2 + 3^2}} = \sqrt{13}. \] Solving this gives two possibilities for the equation. Step 3: Solve for the coordinates of \( P \). We solve the system of equations for both cases and find that the coordinates of \( P \) are \( (1, -6) \). \bigskip

Answer 2

Given:
- Point \( P \) lies on the line:
\[ x + y + 5 = 0 \] - The perpendicular distance from \( P \) to the line:
\[ 2x + 3y + 3 = 0 \] is \( \sqrt{13} \).

Step 1: Let \( P = (x, y) \) satisfy:
\[ x + y + 5 = 0 \implies y = -x - 5 \]

Step 2: Distance of point \( (x, y) \) from line \( 2x + 3y + 3 = 0 \) is:
\[ d = \frac{|2x + 3y + 3|}{\sqrt{2^2 + 3^2}} = \frac{|2x + 3y + 3|}{\sqrt{13}} = \sqrt{13} \] Multiply both sides by \( \sqrt{13} \):
\[ |2x + 3y + 3| = 13 \]

Step 3: Substitute \( y = -x - 5 \):
\[ |2x + 3(-x - 5) + 3| = 13 \] \[ |2x - 3x - 15 + 3| = 13 \] \[ |-x - 12| = 13 \] \[ |x + 12| = 13 \] So:
\[ x + 12 = \pm 13 \] Two cases:
1) \( x + 12 = 13 \implies x = 1 \)
2) \( x + 12 = -13 \implies x = -25 \)

Step 4: Find corresponding \( y \):
For \( x = 1 \):
\[ y = -1 - 5 = -6 \] For \( x = -25 \):
\[ y = 25 - 5 = 20 \]

Step 5: So, possible points are \( (1, -6) \) and \( (-25, 20) \).

Step 6: Check which is correct (both satisfy conditions, but generally first is preferred).

Therefore,
\[ \boxed{(1, -6)} \]