If y $$=$$ mx $$+$$ $$4$$ is a tangent to both the parabolas $$y2=4x$$ and $$x2=2by$$, then b is equal to

Question

If y $$=$$ mx $$+$$ $$4$$ is a tangent to both the parabolas $$y2=4x$$ and  $$x2=2by$$, then b is equal to

Infinity Learn · Accepted Answer

The given line equation is y $$=$$ $$mx+4$$                       ….(i) The equation of the tangent to the parabola $$y2=4x$$ is   $$y=mx+1m$$                        ….(ii)from$$ $$(i) and (ii) $$4=1m$$⇒$$m=14So$$ line $$y=14x+4$$ is also tangent to parabola $$x2=2by$$ , so $$solvex2=2bx+164$$, it must have only one solution, it implies that discriminant must be zero. ⇒$$2x2$$−bx−$$16b=0$$  ⇒$$D=0$$⇒$$b2$$−$$4$$×$$2$$×−$$16b=0$$     ⇒$$b2+32$$×$$4b=0$$    $$b=$$−$$128$$,$$b=0(not$$ $$possible)$$

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

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If y = mx + 4 is a tangent to both the parabolas y2=4x and x2=2by, then b is equal to

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