Question

If
the lines 3 + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then find the value of k​

Answer 1

Correct Question :

If the lines 3x+2ky=2 and 2x+5y+1=0 are parallel, then find the value of k

Theory :

The system of equations

$\sf\:a_1x+b_1y+c_1=0....(1)$

and $\sf\:a_2x+b_2y+c_2=0...(2)$

1. Intersecting , if $\sf\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}$
2. Parallel , if $\sf\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}$
3. Coincident, if $\sf\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$

Solution :

The given system of equations :

$\sf\:3x+2ky-2=0$

and $\sf\:2x+5y+1=0$

This system of equation is of the form

$\sf\:a_1x+b_1y+c_1=0$

and $\sf\:a_2x+b_2y+c_2=0$

Where , $\sf\:a_1=3\:\:b_1=2k\:\:c_1=-2$

and $\sf\:a_2=2\:\:b_2=5\:\:c_2=1$

For Parallel lines :

$\sf\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}$

Put the values , then

$\sf\implies\dfrac{3}{2}=\dfrac{2k}{5}\neq\dfrac{(-2)}{1}$

$\sf\implies\dfrac{3}{2}=\dfrac{2k}{5}$

$\sf\implies\:k=\dfrac{3\times5}{2\times2}$

$\sf\implies\:k=\dfrac{15}{4}$

Therefore , the value of k is 15/4

Answer 2

Answer:

Answer is 15/4= 3.75

Step-by-step explanation:

a1 X+ b1 Y+ c1 = 0

a2 X + b2 Y+ c2 = 0

from the question a1= 3, b1 =2k, c1 = -2

a2 = 2, b2 = 5, c2 = 1

If two lines are parallel then the following condition must follows :-

a1 / a2 = b1 / b2 # (Not equal) c1/c2

a1/a2 = 3/2

b1/b2 = 2k /5

by cross multiplication we get

3*5= 2k * 2

15=4k

k=15/4

k=3.75