Question

The chord of contact of tangents from three points $A,B,C$ to the circle $x^2+y^2=a^2$ are concurrent, then A, B, C will

• A. Be concyclic
• B. Be collinear
• C. Form the vertices of a triangle
• D. None of these

Answer 1

The correct option is B Be collinear

Let the coordinates of A, B and C be $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$, respectively. Then, the chords of contact of tangents drawn from A, B and C are
$x{x}_1+y{y}_1=a^2,x{x}_2+y{y}_2=a^2$ and $x{x}_3+y{y}_3=a^2$, respectively. These three lines will be concurrent, if
$\left|\begin{array}{ccc}{x}_1& y_1& -a^2\\ x_2& y_2& -a^2\\ x_3& y_3& -a^2\end{array}\right|=0$
$\Rightarrow a^2\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$
$\Rightarrow$ Points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ are collinear.