Respuesta :
If two lines are perpendicular, then their slopes are opposite reciprocals.
This means that if you consider the lines:
[tex]y_1=m_1x_1+b_1[/tex][tex]y_2=m_2x_2+b_2[/tex]That are perpendicular, the relationship between their slopes is the following:
[tex]m_2=-\frac{1}{m_1}[/tex]To determine the equation of a line perpendicular to y-2=4(x-1), the first step is to determine the value of its slope. This equation is given in the point-slope form which has the following structure:
[tex]y-y_1=m(x-x_1)[/tex]Where
m represents the slope of the line
(x₁,y₁) represent the coordinates of one point of the line.
On the given equation, the slope is the coefficient that multiplies the parentheses term, m=4
We know that the slope of a line perpendicular to the given line will be the inverse opposite of m=4, then the slope of the perpendicular line will be:
[tex]m=-\frac{1}{4}[/tex]Using the coordinates of the given point (-4,5), the slope m=-1/4, and the point-slope form, you can determine the equation as follows:
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y-5=-\frac{1}{4}(x-(-4)) \\ y-5=-\frac{1}{4}(x+4) \end{gathered}[/tex]So, the equation of the line, that is perpendicular to y-2=4(x-1) and passes through the point (-4,5) is
[tex]y-5=-\frac{1}{4}(x+4)[/tex]