Question

An equilateral triangle is inscribed in the circle $x^2+y^2=a^2$ with the vertex at $(a,0)$ . The equation of the side opposite to this vertex is:

• A. $2x-a=0$
• B. $x+a=0$
• C. $2x+a=0$
• D. $3x-2a=0$

Answer 1

The correct option is C $2x+a=0$

Given equation of circle is $x^2+y^2=a^2$
Center is at (0,0) and radius =a
Now , given vertex of equilateral triangle inscribed in the circle is (a,0).
We know that circumcenter and centroid coincides in an equilateral triangle .
So, centroid is (0,0)
Let equation of BC is x=h
Let M(h,k) be the mid-point of BC.
So, the coordinates of B and C are $(h,\sqrt{a^2-h^2})$ and $(h,-\sqrt{a^2-h^2})$ repectively.
$\frac{h+h+a}{3}=0$
$\Rightarrow 2h+a=0$
$\Rightarrow h=-\frac{a}{2}$
Equation of side BC is $x=-\frac{a}{2}$
or, $2x+a=0$