If the lines $3 x + b y + 5 = 0$ and $a x - 5 y + 7 = 0$ are perpendicular to each other, find the relation connecting $a$ and $b$ .

Open in App

Solution

Step1: Calculation of slope of line $3 x + b y + 5 = 0$ .
The slope-intercept form of the equation of a line is given by $y = m x + c$ , where $m$ is the slope and $c$ is the $Y$ -intercept.
Simplify the equation $3 x + b y + 5 = 0$ to convert it into slope-intercept form.
$3 x + b y + 5 = 0 \Rightarrow b y = - 3 x - 5 (s u b t r a c t i n g 3 x + 5 b o t h s i d e) ∴ y = \frac{- 3}{b} x - \frac{5}{b} (d i v i d i n g b o t h s i d e b y b) - e q u a t i o n (1)$
Comparing equation (1) with equation $y = m x + c$ , we get the slope of the line $3 x + b y + 5 = 0$ as $m_{1} = - \frac{3}{b}$ .
Step2: Calculation of slope of line $a x - 5 y + 7 = 0$ .
Simplify the equation $a x - 5 y + 7 = 0$ to convert it into slope-intercept form.
$a x - 5 y + 7 = 0 \Rightarrow 5 y = a x + 7 (a d d i n g 5 y t o b o t h s i d e s a n d r e a r r a n g i n g) ∴ y = \frac{a}{5} x + \frac{7}{5} (d i v i d i n g b o t h s i d e s b y 5) - e q u a t i o n (2)$
Comparing equation (2) with equation $y = m x + c$ , we get the slope of the line $a x - 5 y + 7 = 0$ as $m_{2} = \frac{a}{5}$ .
Step3: Find the relation between $a$ and $b$ .
Since, the lines $3 x + b y + 5 = 0$ and $a x - 5 y + 7 = 0$ are perpendicular, the product of their slopes will be equal to $- 1$ .
$\Rightarrow m_{1} \cdot m_{2} = - 1 \Rightarrow - \frac{3}{b} . \frac{a}{5} = - 1 \Rightarrow - \frac{3 a}{5 b} = - 1 \Rightarrow - 3 a = - 5 b (m u l t i p l y i n g b o t h s i d e b y 5 b) ∴ 3 a = 5 b (c a n c e l l i n g o u t t h e n e g a t i v e s i g n)$
Hence, the relation connecting $a$ and $b$ is $3 a = 5 b$ .

Suggest Corrections

Similar questions

(i) Lines 2x - by + 5 = 0 and ax + 3y = 2 are parallel to each other. Find the relation connecting a and b.

(ii) Lines mx + 3y + 7 = 0 and 5x - ny - 3 = 0 are perpendicular to each other. Find the relation connecting m and n.

Find the value of k for which the lines kx - 5y + 4 = 0 and 5x - 2y + 5 = 0 are perpendicular to each other

(i) Lines 2x - by + 5 = 0 and ax + 3y = 2 are parallel to each other. Find the relation connecting a and b.

(ii) Lines mx + 3y + 7 = 0 and 5x - ny - 3 = 0 are perpendicular to each other. Find the relation connecting m and n.

2 x - b y + 5 = 0

a x + 3 y = 2

a

b

Lines $2 x - b y + 5 = 0$ and $a x + 3 y = 2$ are parallel. Find the relation connecting $a$ and $b$ .

Join BYJU'S Learning Program