Question

If the chord of contact of tangent from a point P to a given circle passes through Q, then the circle on PQ as diameter

• A. cuts the given circle orthogonally
• B. touches the given circle externally
• C. touches the given circle internally
• D. None of these

Answer 1

The correct option is A cuts the given circle orthogonally

let the circle equation be $x^2+y^2+2gx+2fy+c=0$(equation 1)
let the point P be$(x_1,y_1)$.
equation of chord of contact is $x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$
let the point Q be $(x_2,y_2)$.
since the chord of contact line passes through the point Q,subsitute the point Q in the chord of contact equation.
$x_1{x}_2+y_1{y}_2+g(x_1+x_2)+f(y_1+y_2)+c=0$ (equation 3)
equation of circle passing through the points P and Q as end points of diameter is
$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$.(equation 2)
$x^2+y^2+2gx+2fy+c=0x^2+y^2+2g_{1}x+{2f}_{1}y+c_1=0$
condition for orthogonality of two circle equations is
$2g{g}_1+2f{f}_1=c+c_1.$
for the assumed circle and the circle passing through the points P and Q as end point's of diameter the orthogonality condition satisfies.(for equation 1 &2).
we get $-g(x+x_1)-f(y+y_1)=c+x{x}_1+y{y}_1$
which is equal to equation 3.