Answer
a. $k=\frac{-1}{2}$
b. $k=\frac{-7}{2}$
Work Step by Step
Let $(x_1,y_1)=(4,-1)$
$(x_2,y_2)=(k,2)$
a. Parallel to $2x+3y=6$
By writing the equation in slope intercept form
$2x+3y=6$
$y=-\frac{2}{3}x+2$
The slope is $-\frac{2}{3}$
Since the lines are parallel:
$\frac{2-(-1)}{k-4}=-\frac{2}{3}$
$-2k+8=9$
$k=\frac{-1}{2}$
b. Perpendicular to $5x-2y=-1$
By writing the equation in slope intercept form
$5x-2y=-1$
$y=\frac{5}{2}x+\frac{1}{2}$
The slope is $\frac{5}{2}$
Since the lines are perpendicular,
$m_1.m=-1$
$\frac{5}{2}m=-1$
$m=\frac{-2}{5}$
Then, $\frac{-2}{5}=\frac{2-(-1)}{k-4}$
$-2k+8=15$
$k=\frac{-7}{2}$
