The common tangent to the parabolas y^2=32x a | vedclass

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(A) The given parabolas are $y^2=32x$ and $x^2=256y$.The equation of a common tangent to the parabolas $y^2=4ax$ and $x^2=4by$ is given by $b^{1/3}y +

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$2x+4y+64=0$

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(A) The given parabolas are $y^2=32x$ and $x^2=256y$.The equation of a common tangent to the parabolas $y^2=4ax$ and $x^2=4by$ is given by $b^{1/3}y + a^{1/3}x + (a^2b^2)^{1/3} = 0$.Comparing $y^2=32x$ with $y^2=4ax$,we get $4a=32 \Rightarrow a=8$.Comparing $x^2=256y$ with $x^2=4by$,we get $4b=256 \Rightarrow b=64$.Substituting these values into the formula:$(64)^{1/3}y + (8)^{1/3}x + (8^2 	imes 64^2)^{1/3} = 0$$4y + 2x + (64 	imes 4096)^{1/3} = 0$$4y + 2x + (262144)^{1/3} = 0$$4y + 2x + 64 = 0$Dividing by $2$,we get $x + 2y + 32 = 0$. However,checking the options,the equation $2x+4y+64=0$ is equivalent to $x+2y+32=0$.

The common tangent to the parabolas $y^2=32x$ and $x^2=256y$ is:

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