If the chord of contact of tangents from a po | vedclass

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(D) The equation of the chord of contact of tangents from $P(x_1, y_1)$ to the parabola $y^2 = 4ax$ is given by $yy_1 = 2a(x + x_1)$.This can be rewri

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hyperbola

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(D) The equation of the chord of contact of tangents from $P(x_1, y_1)$ to the parabola $y^2 = 4ax$ is given by $yy_1 = 2a(x + x_1)$.This can be rewritten as $x = \frac{y_1}{2a}y - x_1$.Since this line touches the parabola $x^2 = 4by$,we substitute $x$ in the parabola equation:$(\frac{y_1}{2a}y - x_1)^2 = 4by$$\frac{y_1^2}{4a^2}y^2 - \frac{x_1y_1}{a}y + x_1^2 = 4by$$\frac{y_1^2}{4a^2}y^2 - (\frac{x_1y_1}{a} + 4b)y + x_1^2 = 0$.For the line to be a tangent,the discriminant $D$ must be zero:$D = (\frac{x_1y_1}{a} + 4b)^2 - 4(\frac{y_1^2}{4a^2})(x_1^2) = 0$$\frac{x_1^2y_1^2}{a^2} + \frac{8bx_1y_1}{a} + 16b^2 - \frac{x_1^2y_1^2}{a^2} = 0$$\frac{8bx_1y_1}{a} + 16b^2 = 0$$x_1y_1 + 2ab = 0$.Replacing $(x_1, y_1)$ with $(x, y)$,the locus is $xy = -2ab$,which represents a hyperbola.

If the chord of contact of tangents from a point $P(x_1, y_1)$ to the parabola $y^2 = 4ax$ touches the parabola $x^2 = 4by$,the locus of $P$ is:

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