Question

Line k has the equation y = -2 - 10.Line lis perpendicular to line k, and passes through the point (3,2).Find an equation for line l in both slope-intercept form and point-slope form using the given point.An equation for l in slope-intercept form is:An equation for { in point-slope form is:.

Answer 1

The equation of line 'k' is given as,

$y=-x-10$

Consider that the slope-intercept form of a line having slope 'm' and y-intercept 'c' is given by,

$y=mx+c$

On comparing, it can be claimed that the slope of line 'k' is,

$m_k=-1$

Theorem: The product of slopes of two perpendicular lines is always -1.

Given that line 'l' is perpendicular to line 'k',

$\begin{gathered} m_l\cdot m_k=-1 \\ m_l\cdot(-1)=-1 \\ m_l=1 \end{gathered}$

Then the equation of the line 'l' considering the y-intercept as 'C' is obtained as,

$\begin{gathered} y=m_kx+C \\ y=x+C \end{gathered}$

Given that the line 'l' passes through the point (3,2),

$\begin{gathered} 2=3+C \\ C=2-3 \\ C=-1 \end{gathered}$

So the equation of the line 'l' in the slope-intercept form will be,

$y=x-1$

Consider that the point-slope form of a line is given by,

$y-y_1=m\cdot(x-x_1)$

As discussed above, the slope of the line 'l' is 1, and it passes through the point (2,3),

$\begin{gathered} m=1_{} \\ (x_1,y_1)=(3,2) \end{gathered}$

Then the corresponding equation of line 'l' in point-slope form, will be,

$y-2=1\cdot(x-3)$

Thus, the equation of line 'l' in slope-intercept form and point-slope form respectively,

$\begin{gathered} y=x-1 \\ y-2=1\cdot(x-3) \end{gathered}$