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Home » Find the conditions that the straight lines y = m1x + c1, y = m2x + c2 and y = m3x + c3 may meet in a point.
Find the conditions that the straight lines y = m1x + c1, y = m2x + c2 and y = m3x + c3 may meet in a point.
CBSE | Class 11 | Exercise 23.11 | Maths | RD Sharma | Straight Lines
Find the conditions that the straight lines y = m1x + c1, y = m2x + c2 and y = m3x + c3 may meet in a point.

Given:

    \[{{m}_{1}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{1}}~=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\]

    \[{{m}_{2}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{2}}~=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\]

and,

    \[{{m}_{3}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{3}}~=\text{ }0\text{ }\ldots \text{ }\left( 3 \right)\]

since, the three lines are concurrent.

Now,

    \[{{m}_{1}}\left( -{{c}_{3}}~+\text{ }{{c}_{2}} \right)\text{ }+\text{ }1\left( {{m}_{2}}{{c}_{3}}-{{m}_{3}}{{c}_{2}} \right)\text{ }+\text{ }{{c}_{1}}\left( -{{m}_{2}}~+\text{ }{{m}_{3}} \right)\text{ }=\text{ }0\]

    \[{{m}_{1}}\left( {{c}_{2}}-{{c}_{3}} \right)\text{ }+\text{ }{{m}_{2}}\left( {{c}_{3}}-{{c}_{1}} \right)\text{ }+\text{ }{{m}_{3}}\left( {{c}_{1}}-{{c}_{2}} \right)\text{ }=\text{ }0\]

∴ The required condition is:

    \[{{m}_{1}}\left( {{c}_{2}}-{{c}_{3}} \right)\text{ }+\text{ }{{m}_{2}}\left( {{c}_{3}}-{{c}_{1}} \right)\text{ }+\text{ }{{m}_{3}}\left( {{c}_{1}}-{{c}_{2}} \right)\text{ }=\text{ }0\]

i

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