The pole of the line lx + my + n = 0 with res | vedclass

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(A) The equation of the polar of a point $P(x_{1}, y_{1})$ with respect to the parabola $y^{2} = 4ax$ is given by $yy_{1} = 2a(x + x_{1})$.This can be

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$\left(\frac{n}{l}, \frac{-2am}{l}\right)$

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(A) The equation of the polar of a point $P(x_{1}, y_{1})$ with respect to the parabola $y^{2} = 4ax$ is given by $yy_{1} = 2a(x + x_{1})$.This can be rewritten as $2ax - yy_{1} + 2ax_{1} = 0 \dots(i)$.The given line is $lx + my + n = 0 \dots(ii)$.Comparing equations $(i)$ and $(ii)$,we have:$\frac{2a}{l} = \frac{-y_{1}}{m} = \frac{2ax_{1}}{n}$.From the first and third ratios: $\frac{2a}{l} = \frac{2ax_{1}}{n} \Rightarrow x_{1} = \frac{n}{l}$.From the first and second ratios: $\frac{2a}{l} = \frac{-y_{1}}{m} \Rightarrow y_{1} = \frac{-2am}{l}$.Thus,the pole is $\left(\frac{n}{l}, \frac{-2am}{l}\right)$.

List-$I$	List-$II$
$(A)$ Focus	$(I)$ $(4,2)$
$(B)$ Vertex	$(II)$ $(3,2)$
$(C)$ One end of the latus rectum	$(III)$ $(2,3)$
$(D)$ Point of intersection of the axis and directrix	$(IV)$ $(2,4)$
	$(V)$ $(2,2)$

The pole of the line $lx + my + n = 0$ with respect to the parabola $y^{2} = 4ax$ is

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