- Home
- Standard 11
- Mathematics
જો $(a_1, b_1)$ અને $(a_2, b_2)$ બિંદુથી સમાન અંતરે આવેલા બિંદુના બિંદુપથનું સમીકરણ $(a_1 - a_2)x + (b_1 - b_2)y + c = 0$ હોય, તો $'c'$ નું મૂલ્ય શોધો ?
$\sqrt {a_1^2\,\, + \,\,b_1^2\,\, - \,\,a_2^2\,\, - \,\,b_2^2} $
$\frac{1}{2}\,\,\,(a_2^2\,\, + \,\,b_2^2\,\, + \,\,a_1^2\,\, - \,\,b_1^2\,)$
$a_1^2\,\, - \,a_2^2\,\, + \,\,b_1^2\,\, - \,\,b_2^2$
$\frac{1}{2}\,\,(a_1^2\,\, + \,\,a_2^2\,\, + \,\,b_1^2\,\, + \,\,b_1^2)$
Solution
Let $( h , k )$ be the point on the locus.
Then by the given conditions,
$\left(h-a_1\right)^2+\left(k-b_1\right)^2=\left(h-a_2\right)^2+\left(k-b_2\right)^2$
$\Rightarrow 2 h\left(a_1-a_2\right)+2 k\left(b_1-b_2\right)+a_2^2-a_1^2+b_2^2-b_1^2=0$
$\Rightarrow h\left(a_1-a_2\right)+k\left(b_1-b_2\right)+\frac{1}{2}\left(a_2^2+b_2^2-a_1^2-b_1^2\right)=0$
$A l s o$, since $(h, k)$ lies on the given locus, therefore
$\left(a_1-a_2\right) x+\left(b_1-b_2\right) y+c=0$
Comparing Eqs.$(i)$ and $(ii)$, we get
$c=\frac{1}{2}\left(a_2^2+b_2^2-a_1^2-b_1^2\right)$
