Question

How do you write an equation of a line passing through $(3,2)$, perpendicular to $y = 5x + 2$ ?

Answer 1

Use point-slope formula to write the equation of the line passing through the given point and having the slope of the given equation of line. Compare the equation of line to the general equation of line and find the value of slope. Use that value of slope and write slope of perpendicular. Use that value of slope to form a new equation.

Complete step by step solution:
We are given a line $y = 5x + 2$
Compare the given equation of line with general slope intercept form of line i.e. $y = mx + c$
We get the value of slope i.e. $m = 5$
Now we have to find the equation of the line that is perpendicular to the given line. Since we know for a line having slope ‘m’, the slope of line perpendicular to the line will be $\dfrac{{ - 1}}{m}$; then the slope for the perpendicular will be:
$m' = \dfrac{{ - 1}}{m} = \dfrac{{ - 1}}{5}$
We know that the equation of line having slope m and passing through the point $({x_1},{y_1})$ can be written using the point slope formula i.e. $y - {y_1} = m'(x - {x_1})$
Comparing the values given in the question we get $m' = \dfrac{{ - 1}}{5}$ and point $({x_1},{y_1}) = (3,2)$
Substitute the value of $m' = \dfrac{{ - 1}}{5}$ and ${x_1} = 3;{y_1} = 2$ in the point slope formula
$\Rightarrow y - 2 = \dfrac{{ - 1}}{5} \times (x - 3)$
Cross multiply the denominator from right hand side to left hand side of the equation
$\Rightarrow 5y - 10 = - x + 3$
Bring all terms to left hand side of the equation
$\Rightarrow x + 5y - 10 - 3 = 0$
$\Rightarrow x + 5y - 13 = 0$
$\therefore$ The equation of a line passing through $(3,2)$, perpendicular to $y = 5x + 2$ is $x + 5y - 13 = 0$

Note: Many students make the mistake of writing the slope of line perpendicular to a given line as same as the slope of the line which is wrong. Keep in mind slope of perpendicular will be negative if the reciprocal of the given slope of a line as perpendicular of a line is not in the same direction, it is exactly making the right angle with the given line.