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Home » On comparing the ratios a1/a2 , b1/b2 , c1/c2 find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
On comparing the ratios a1/a2 , b1/b2 , c1/c2 find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
CBSE | Class 10 | Exercise 3.2 | Guides | Pair of Linear Equations in Two Variables
On comparing the ratios a1/a2 , b1/b2 , c1/c2 find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(i) 5x – 4y + 8 = 0

7x + 6y – 9 = 0

(ii) 9x + 3y + 12 = 0

18x + 6y + 24 = 0

(iii) 6x – 3y + 10 = 0

2x – y + 9 = 0

Arrangements:

(I) Given articulations;

   

    \[5x\text{ }\text{ }4y\text{ }+\text{ }8\text{ }=\text{ }0\]

    

    \[7x\text{ }+\text{ }6y\text{ }\text{ }9\text{ }=\text{ }0\]

Contrasting these conditions and

    \[a1x+b1y+c1\text{ }=\text{ }0\]

 

Also,

    \[a2x+b2y+c2\text{ }=\text{ }0\]

 

We get,

    \[a1\text{ }=\text{ }5,\text{ }b1\text{ }=\text{ }-\text{ }4,\text{ }c1\text{ }=\text{ }8\]

 

    \[a2\text{ }=\text{ }7,\text{ }b2\text{ }=\text{ }6,\text{ }c2\text{ }=\text{ }-\text{ }9\]

 

    \[\left( a1/a2 \right)\text{ }=\text{ }5/7\]

 

    \[\left( b1/b2 \right)\text{ }=\text{ }-\text{ }4/6\text{ }=\text{ }-\text{ }2/3\]

    \[\left( c1/c2 \right)\text{ }=\text{ }8/\text{ }-\text{ }9\]

Since,

    \[\left( a1/a2 \right)\text{ }\ne \text{ }\left( b1/b2 \right)\]

 

Along these lines, the sets of conditions given in the inquiry have a one of a kind arrangement and the lines cross each other at precisely one point.

(ii) Given articulations:

     

    \[9x\text{ }+\text{ }3y\text{ }+\text{ }12\text{ }=\text{ }0\]

 

     

    \[18x\text{ }+\text{ }6y\text{ }+\text{ }24\text{ }=\text{ }0\]

Contrasting these conditions and

    \[a1x+b1y+c1\text{ }=\text{ }0\]

 

What’s more,

    \[a2x+b2y+c2\text{ }=\text{ }0\]

 

We get,

    \[a1\text{ }=\text{ }9,\text{ }b1\text{ }=\text{ }3,\text{ }c1\text{ }=\text{ }12\]

 

    \[a2\text{ }=\text{ }18,\text{ }b2\text{ }=\text{ }6,\text{ }c2\text{ }=\text{ }24\]

 

    \[\left( a1/a2 \right)\text{ }=\text{ }9/18\text{ }=\text{ }1/2\]

 

    \[\left( b1/b2 \right)\text{ }=\text{ }3/6\text{ }=\text{ }1/2\]

 

    \[\left( c1/c2 \right)\text{ }=\text{ }12/24\text{ }=\text{ }1/2\]

 

Since

    \[\left( a1/a2 \right)\text{ }=\text{ }\left( b1/b2 \right)\text{ }=\text{ }\left( c1/c2 \right)\]

 

Thus, the sets of conditions given in the inquiry have boundless potential arrangements and the lines are incidental.

(iii) Given Expressions;

      

    \[6x\text{ }\text{ }3y\text{ }+\text{ }10\text{ }=\text{ }0\]

 

      

    \[2x\text{ }\text{ }y\text{ }+\text{ }9\text{ }=\text{ }0\]

Contrasting these conditions and

    \[a1x+b1y+c1\text{ }=\text{ }0\]

What’s more,

    \[a2x+b2y+c2\text{ }=\text{ }0\]

We get,

    \[a1\text{ }=\text{ }6,\text{ }b1\text{ }=\text{ }-\text{ }3,\text{ }c1\text{ }=\text{ }10\]

 

    \[a2\text{ }=\text{ }2,\text{ }b2\text{ }=\text{ }-\text{ }1,\text{ }c2\text{ }=\text{ }9\]

    \[\left( a1/a2 \right)\text{ }=\text{ }6/2\text{ }=\text{ }3/1\]

    \[\left( b1/b2 \right)\text{ }=\text{ }-\text{ }3/\text{ }-\text{ }1\text{ }=\text{ }3/1\]

    \[\left( c1/c2 \right)\text{ }=\text{ }10/9\]

 

Since

    \[\left( a1/a2 \right)\text{ }=\text{ }\left( b1/b2 \right)\text{ }\ne \text{ }\left( c1/c2 \right)\]

  Along these lines, the sets of conditions given in the inquiry are corresponding to one another and the lines never cross each other anytime and there is no conceivable answer for the given pair of conditions.

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