0

I was trying to understand a solution, when a encountered this line:

$$or,\;\sum(a+\lambda l)^2=\lambda^2(l^2+m^2+n^2)\\[10pt]or,\;\lambda=-\frac{a^2+b^2+c^2}{2(al+bm+cn)}$$

I tried various method to reach the second equation from the first one, I couldn't. Here is the image for reference. [This is not a homework question, it's purely an evaluation based problem/technique]. I tried, taking square root and all sorts of elementary algebra techniques, didn't work.

Captain Lama
  • 25,743
peaceHoper
  • 139

1 Answers1

1

Hint: The expression $\ \sum (a+\lambda l)^2\ $ is just very lazy shorthand notation for $$ (a+\lambda l)^2 + (b+\lambda m)^2 + (c+\lambda n)^2\ . $$ If you expand this expression out, your second equation should fall into your lap.

Note, however, that there was a minus sign missing from the right hand side of your second equation, which I have now added.

lonza leggiera
  • 28,646
  • 2
  • 12
  • 33
  • I have done it...it starts getting too lengthy , then I get stuck. – peaceHoper May 21 '20 at 12:19
  • 1
    Too lengthy? There are only 9 terms in the expansion, and three of them are $\ \lambda^2l^2+ \lambda^2m^2+ \lambda^2n^2\ $, which cancel out those same terms on the right hand side of your first equation. – lonza leggiera May 21 '20 at 13:25