4 A line passos through \( (1,1) \) and \( (3,5) \). ind wein Ard the equation of a line perpendicular to this ine passing through \( (2,-1) \)

To find the equation of the line that passes through \( (1,1) \) and \( (3,5) \), first, we calculate the slope of the original line. The slope \( m \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 1}{3 - 1} = \frac{4}{2} = 2 \] The slope of the line perpendicular to this one is the negative reciprocal, which is: \[ m_{\text{perpendicular}} = -\frac{1}{2} \] Next, we can use the point-slope form of the equation of a line, which is: \[ y - y_1 = m(x - x_1) \] Substituting in the point \( (2,-1) \) and the perpendicular slope: \[ y - (-1) = -\frac{1}{2}(x - 2) \] This simplifies to: \[ y + 1 = -\frac{1}{2}x + 1 \] Subtracting 1 from both sides gives us: \[ y = -\frac{1}{2}x \] So, the equation of the line that is perpendicular to the original line and passes through \( (2,-1) \) is: \[ y = -\frac{1}{2}x \]