44. The perpendicular distance from origin to the line \( \frac{x+2}{-1}=\frac{y+1}{2}=1 \) \( \begin{array}{llll}\text { (A) } \frac{15}{14} & \text { (B) } \sqrt{\frac{531}{14}} & \text { (C) } \frac{531}{14}\end{array} \)

To find the perpendicular distance from the origin to the line given by the equation \( \frac{x+2}{-1}=\frac{y+1}{2}=1 \), we first need to convert this symmetric form into the standard linear equation form \( Ax + By + C = 0 \). ### Step 1: Convert the symmetric equation to standard form From the equation \( \frac{x+2}{-1} = 1 \), we can express \( x \): \[ x + 2 = -1 \implies x = -3 \] From the equation \( \frac{y+1}{2} = 1 \), we can express \( y \): \[ y + 1 = 2 \implies y = 1 \] Now, we can find the parametric equations for the line: - For \( t = 1 \): - \( x = -3 \) - \( y = 1 \) Next, we can express the line in terms of a point and a direction vector. The direction vector can be derived from the coefficients in the symmetric equation: - The direction vector is \( (-1, 2) \). Using the point \((-3, 1)\) and the direction vector \((-1, 2)\), we can write the parametric equations: \[ x = -3 - t, \quad y = 1 + 2t \] ### Step 2: Find the standard form of the line To find the standard form, we can eliminate \( t \): 1. From \( x = -3 - t \), we have \( t = -3 - x \). 2. Substitute \( t \) into the equation for \( y \): \[ y = 1 + 2(-3 - x) = 1 - 6 - 2x = -2x - 5 \] Rearranging gives us: \[ 2x + y + 5 = 0 \] Thus, the line in standard form is: \[ 2x + y + 5 = 0 \] ### Step 3: Calculate the perpendicular distance from the origin to the line The formula for the distance \( d \) from a point \((x_0, y_0)\) to the line \( Ax + By + C = 0 \) is given by: \[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \] For the line \( 2x + y + 5 = 0 \): - \( A = 2 \) - \( B = 1 \) - \( C = 5 \) Using the origin \((0, 0)\): \[ d = \frac{|2(0) + 1(0) + 5|}{\sqrt{2^2 + 1^2}} = \frac{|5|}{\sqrt{4 + 1}} = \frac{5}{\sqrt{5}} = \sqrt{5} \] ### Step 4: Compare with the options Now we need to check if \( \sqrt{5} \) matches any of the given options: - (A) \( \frac{15}{14} \) - (B) \( \sqrt{\frac{531}{14}} \) - (C) \( \frac{531}{14} \) Calculating \( \sqrt{\frac{531}{14}} \): \[ \sqrt{\frac{531}{14}} = \frac{\sqrt{531}}{\sqrt{14}} \] To find if \( \sqrt{5} \) is equal to any of these, we can square \( \sqrt{5} \): \[ 5 = \frac{531}{14} \implies 5 \cdot 14 = 531 \implies 70 = 531 \quad \text{(not true)} \] Next, we can check \( \frac{15}{14} \): \[ \left(\frac{15}{14}\right)^2 = \frac{225}{196} \quad \text{(not equal to 5)} \] Finally, we can check \( \sqrt{\frac{531}{14}} \): \[ \frac{531}{14} \approx 37.93 \quad \text{(not equal to 5)} \] ### Conclusion None of the options match \( \sqrt{5} \). However, the calculations show that the distance from the origin to the line is indeed \( \sqrt{5} \). Thus, the answer is not among the provided options.