If three lines whose equations are y=m1 x+c1, y=m2 x+c2 and y=m3 x+c3, are concurrent, then I. m1(c2-c3)+m2(c3-c1)+m3(c1-c2)=0 II. m1(c1-c2)+m2(c2-c3)+m3(c3-c1)=0

Question

If three lines whose equations are y=m1 x+c1, y=m2 x+c2 and y=m3 x+c3, are concurrent, then I. m1(c2-c3)+m2(c3-c1)+m3(c1-c2)=0 II. m1(c1-c2)+m2(c2-c3)+m3(c3-c1)=0 (A) Both I and II are true (B) Only l is true (C) Only II is true (D) Both I and II are false

Tardigrade Admin · Accepted Answer

Given equation of lines are y=m1 x+c1 .....(i) y=m2 x+c2 .....(ii) y=m3 x+c3.....(iii) On solving Eqs. (i) and (ii), we get m1 x+c1=m2 x+c2 ⇒ m1 x-m2 x=c2-c1 ⇒ x=(c2-c1/m1-m2) From Eq. (i), y=m1((c2-c1/m1-m2))+(c1/1) ⇒ y=(m1 c2-m1 c1+m1 c1-m2 c1/m1-m2) ⇒ y=(m1 c2-m2 c1/m1-m2) On putting the values of x and y in Eq. (iii), we get (m1 c2-m2 c1/m1-m2)=m3((c2-c1/m1-m2))+(c3/1) ⇒ (m1 c2-m2 c1/m1-m2)=(m3 c2-m3 c1+m1 c3-m2 c3/m1-m2) ⇒ m1 c2-m2 c1=m3 c2-m3 c1+m1 c3-m2 c3 ⇒ m1 c2-m2 c1-m3 c2+m3 c1-m1 c3+m2 c3=0 ⇒ m1(c2-c3)+m2(c3-c1)+m3(c1-c2)=0 Hence proved.

Q. If three lines whose equations are y=m1​x+c1​, y=m2​x+c2​ and y=m3​x+c3​, are concurrent, then I. m1​(c2​−c3​)+m2​(c3​−c1​)+m3​(c1​−c2​)=0 II. m1​(c1​−c2​)+m2​(c2​−c3​)+m3​(c3​−c1​)=0

Both I and II are true

Only l is true

Only II is true

Both I and II are false

Q. If three lines whose equations are $y = m_{1} x + c_{1}$ , $y = m_{2} x + c_{2}$ and $y = m_{3} x + c_{3}$ , are concurrent, then
I. $m_{1} (c_{2} - c_{3}) + m_{2} (c_{3} - c_{1}) + m_{3} (c_{1} - c_{2}) = 0$
II. $m_{1} (c_{1} - c_{2}) + m_{2} (c_{2} - c_{3}) + m_{3} (c_{3} - c_{1}) = 0$