The equation of the common tangent of the par | vedclass

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(A) Given parabolas are $x^2=108y$ and $y^2=32x$.For $x^2=4ay$,$4a=108 \Rightarrow a=27$. The tangent is $y=mx-am^2$,so $y=mx-27m^2$ $... (i)$.For $y^

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$2x+3y+36=0$

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(A) Given parabolas are $x^2=108y$ and $y^2=32x$.For $x^2=4ay$,$4a=108 \Rightarrow a=27$. The tangent is $y=mx-am^2$,so $y=mx-27m^2$ $... (i)$.For $y^2=4ax$,$4a=32 \Rightarrow a=8$. The tangent is $y=mx+\frac{a}{m}$,so $y=mx+\frac{8}{m}$ $... (ii)$.Since $(i)$ and $(ii)$ represent the same line,$-27m^2 = \frac{8}{m}$.$m^3 = -\frac{8}{27} \Rightarrow m = -\frac{2}{3}$.Substituting $m$ into $(i)$:$y = -\frac{2}{3}x - 27(-\frac{2}{3})^2$$y = -\frac{2}{3}x - 27(\frac{4}{9})$$y = -\frac{2}{3}x - 12$$3y = -2x - 36$$2x + 3y + 36 = 0$.

The equation of the common tangent of the parabolas $x^2=108y$ and $y^2=32x$ is

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