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If the parabolas y^2=4a x and y^2=4c(x-b...

If the parabolas `y^2=4a x` and `y^2=4c(x-b)` have a common normal other than the x-axis `(a , b , c` being distinct positive real numbers), then prove that `b/(a-c)> 2.`

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To prove that \( \frac{b}{a - c} > 2 \) given the parabolas \( y^2 = 4ax \) and \( y^2 = 4c(x - b) \) have a common normal other than the x-axis, we will follow these steps: ### Step 1: Write the equations of the normals The equation of the normal to the first parabola \( y^2 = 4ax \) at a point \( (at^2, 2at) \) is given by: \[ y = mx - 2am - at^3 \] For the second parabola \( y^2 = 4c(x - b) \) at a point \( (c(t')^2 + b, 2ct') \), the equation of the normal is: ...
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