How to represent Appolonims theorem in argand plane?
Subham pal , 9 Years ago
Grade 12th pass
Algebra> How to represent Appolonims theorem in ar...
1 AnswersVikas TU
If O be the mid-point of the side MN of the triangle LMN, then
LM2 + LN2 = 2(LO2 + MO2).
LM2 + LN2 = 2(LO2 + MO2).
Proof: Let us choose origin of rectangular Cartesian co-ordinates at O and x-axis along the side MN and OY as the y – axis . If MN = 2a then the co-ordinates of M and N are (- a, 0) and (a, 0) respectively. Referred to the chosen axes if the co-ordinates of L be (b, c) then
LO2 = (b - 0)2 + (C - 0)2 , [Since, co- ordinates of O are (0, 0)]
= b2 + c2;
MO2 = (- a - 0)2 + (0 – 0)2 = a2
LMB2 = (b + a) 2 + (c – 0)2 = (a + b)2 + c2
And LN2 = (b - a) 2 + (c - 0) 2 = (a - b)2 + c2
Therefore, LM2 + LN2 = (a + b) 2 + c2 + (b - a)2 + c2
= 2(a2 + b2) + 2c2
= 2a2 + 2(b2 + c2)
= 2MO2 + 2LO2
= 2(MO2 + LO2).
= 2(LO2 + MO2). Proved.
LO2 = (b - 0)2 + (C - 0)2 , [Since, co- ordinates of O are (0, 0)]
= b2 + c2;
MO2 = (- a - 0)2 + (0 – 0)2 = a2
LMB2 = (b + a) 2 + (c – 0)2 = (a + b)2 + c2
And LN2 = (b - a) 2 + (c - 0) 2 = (a - b)2 + c2
Therefore, LM2 + LN2 = (a + b) 2 + c2 + (b - a)2 + c2
= 2(a2 + b2) + 2c2
= 2a2 + 2(b2 + c2)
= 2MO2 + 2LO2
= 2(MO2 + LO2).
= 2(LO2 + MO2). Proved.
LIVE ONLINE CLASSES
Prepraring for the competition made easy just by live online class.
Full Live Access
Study Material
Live Doubts Solving
Daily Class Assignments
Other Related Questions on algebra
Can we divide two vectors? If this is possible then how?
Can we divide two vectors? If this is possible then how?
algebra
1 Answer Available
Last Activity: 2 Years ago
How do you write “the product of 18 and q” as an algebraic expression?
How do you write “the product of 18 and q” as an algebraic expression?
algebra
1 Answer Available
Last Activity: 2 Years ago
दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे
दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे
algebra
1 Answer Available
Last Activity: 3 Years ago