Question

without drawing the graph find out whether the lines representing the following pair of linear equations intersect at a point are parallel or coincident 9x-10y=21 & 3/2x-5/3y=7/2

Answer 1

Eq 1 :9x-10y-21=0

Eq 2 : 3/2x -5/2y -7/2=0

=3/2x×6-5/2y×6-7/2×6=0

=3x×3-5y×2-7×3=0

=9x-10y-21=0

a1/a2 = 9/9=1

b1/b2 =-10/-10 =1

c1/c2 =-21/-21 =1

Since a1/a2=b1/b2=c1/c2

These lines are coincident

Answer 2

Conditions to find whether these equations are intersecting, coincident, parallel

Condition for Intersecting:

$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

Condition for Coincident:

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Condition for Parallel:

slope of equation ($1$) = slope of equation ($2$)

$= > m_1 = m_2$

Given,

$9x-10y=21 ----- > (1)$

$\frac{3x}{2}-\frac{5y}{3}=\frac{7}{2} ---- > (2)$

by comparing them with the general line equation $ax+by+c=0$ we get

$a_1 = 9, a_2=\frac{3}{2} \\ \\b_1=(-10), b_2 = \frac{-5}{3} \\ \\c_1 = -21 , c_2=\frac{-7}{2}$

now calculating,

$\frac{a_1}{a_2} = \frac{9}{\frac{3}{2} }\\ \\\frac{a_1}{a_2}= 6$

$\frac{b_1}{b_2}= \frac{-10}{\frac{-5}{3} }\\ \\\frac{b_1}{b_2} = 6$

$\frac{c_1}{c_2} =\frac{-21}{\frac{-7}{2} } \\\frac{c_1}{c_2}= 6$

∴ From the above values we can know that

$\frac{a_1}{a_2} =\frac{b_1}{b_2} = \frac{c_1}{c_2}$

So, the lines given are Coincident in nature.

#SPJ3