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-$II$
$A$. Equation of the tangent drawn at $(2, \sqrt{8})$ on the curve $y^2 = 4x$ is	$(i) -36$
$B$. Equation of the normal to the curve $y^2 = 16x$,that makes an angle of $45^{\circ}$ with its axis is	$(ii) 4$
$C$. The chord joining the points $(x_1, y_1)$ and $(x_2, y_2)$ on the curve $y^2 = 12x$ is a focal chord if $y_1 y_2 =$	$(iii) 8$
$D$. $A$ value of $k$ for which $x - 3 = 0$ is the directrix of the curve $y^2 - kx + 16 = 0$ is	$(iv) x - \sqrt{2}y + 2 = 0$
	$(v) x + y - 12 = 0$
	$(vi) x - y - 12 = 0$

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

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