If y=mx+4 is a tangent to both the parabolas | vedclass

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Question

(C) The line $y=mx+4$ is a tangent to the parabola $y^{2}=4x$. Comparing this with $y=mx+c$,where $c=4$,the condition for tangency $c=\frac{a}{m}$ giv

Vedclass · Accepted Answer

$-128$

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(C) The line $y=mx+4$ is a tangent to the parabola $y^{2}=4x$. Comparing this with $y=mx+c$,where $c=4$,the condition for tangency $c=\frac{a}{m}$ gives $4=\frac{1}{m}$,so $m=\frac{1}{4}$.The line $y=\frac{1}{4}x+4$ is also a tangent to the parabola $x^{2}=2by$. Substituting $y$ from the line equation into the parabola equation,we get $x^{2}=2b(\frac{1}{4}x+4)$.This simplifies to $x^{2}-\frac{b}{2}x-8b=0$.Since the line is tangent,the discriminant $D$ of this quadratic equation must be zero.$D = (-\frac{b}{2})^{2} - 4(1)(-8b) = 0$.$\frac{b^{2}}{4} + 32b = 0$.$b^{2} + 128b = 0$.$b(b+128) = 0$.Since $b 
eq 0$ for a valid parabola,we have $b=-128$.

If $y=mx+4$ is a tangent to both the parabolas $y^{2}=4x$ and $x^{2}=2by$,then $b$ is equal to:

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