Question

Find the equation of the chord of contact of the point (1, 2) with respect to the circle $x^2 + y^2 + 2x + 3y + 1 = 0$.

Answer 1

The equation of the chord of contact is $4x + 7y + 10 = 0$.

Answer 2

The equation of the given circle is $x^2 + y^2 + 2x + 3y + 1 = 0$. Comparing this with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we get $g = 1$, $f = 3/2$, and $c = 1$. The given point is $(x_1, y_1) = (1, 2)$. The equation of the chord of contact is given by $T=0$, where $T = xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c$. Substituting the values: $T = x(1) + y(2) + 1(x+1) + \frac{3}{2}(y+2) + 1$ $T = x + 2y + x + 1 + \frac{3}{2}y + 3 + 1$ $T = 2x + \frac{7}{2}y + 5$ Setting $T=0$: $2x + \frac{7}{2}y + 5 = 0$ Multiplying by 2 to remove the fraction: $4x + 7y + 10 = 0$