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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.`A. `0ltb/(a-c)lt1`B. `b/(a-c)gt2`C. `b/(a-c)lt0`D. `1ltb/(a-c)lt2` |
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Answer» Correct Answer - B The equation of any normal of slope m to the parabola `y^(2)=4c(x-c)` is `y=m(x-b)-2cm-cm^(2)or, y=mx-mb-2cm-cm^(3)` For this to be normal to `y^(2)=4ax`, we must have `-2am-am^(3)=-mb-2cm-cm^(3)` `rArr" "2a+am^(2)=b+2c+cm^(2)` `rArr" "m^(2)=(b+2c-2a)/(a-c)rArrm^(2)=b/(ac)-2rArrm=+-sqrt(b/(a-c)-2)` For m to be real, we must have `-b/(a-c)-2gt0rArr=b/(a-c)gt2` |
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