The condition for the lines lx + my + n = 0 a | vedclass

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(A) Two lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ are conjugate with respect to the circle $x^2 + y^2 = r^2$ if the pole of the first line l

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$r^2(ll_1 + mm_1) = nn_1$

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(A) Two lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ are conjugate with respect to the circle $x^2 + y^2 = r^2$ if the pole of the first line lies on the second line.The pole of the line $lx + my + n = 0$ with respect to the circle $x^2 + y^2 = r^2$ is given by $(x_1, y_1) = (-\frac{lr^2}{n}, -\frac{mr^2}{n})$.Since this point lies on the line $l_1x + m_1y + n_1 = 0$,we have:$l_1(-\frac{lr^2}{n}) + m_1(-\frac{mr^2}{n}) + n_1 = 0$$-l_1lr^2 - m_1mr^2 + n_1n = 0$$nn_1 = r^2(ll_1 + mm_1)$.

The condition for the lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ to be conjugate with respect to the circle $x^2 + y^2 = r^2$ is:

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