Locus of pole of a line with respect to parabola.

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Locus of pole of a line with respect to parabola.

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#1 h Sep 29, 2010, 5:53 PM • 2 Y

Y by Adventure10, Mango247

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$ , are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

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#2 h Oct 3, 2018, 7:27 PM • 2 Y

Y by Adventure10, Mango247

Locus of the poles $M$ is the parabola $P_{2}$ .

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RenheMiResembleRice

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#3 h Jan 22, 2022, 12:50 AM

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vanstraelen wrote:

Locus of the poles $M$ is the parabola $P_{2}$ .

Why? ._.

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#4 h Mar 6, 2022, 9:18 PM

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The tangent line $2\lambda x+y-2p\lambda^{2}=0$ in a point $T(2p\lambda,-2p\lambda^{2})$ of the parabola $x^{2}=-2py$ .

Two points on this tangent line, $P(0,2p\lambda^{2})$ and $Q(p\lambda,0)$ .
The polar lines of these points are $y+2p\lambda^{2}=0$ and $\lambda x-y=0$ , cutting in the pole $M$ of the tangent line.

Eliminating the parameter $\lambda$ , we find $x^{2}+2py=0$ .

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