if-the-radius-of-a-circle-touching-parabola-y-2-4x-at-4-4-and-having-directrix-of-y-2-4x-as-its-normal-is-r-then-find-r-where-x-denote-greatest-integer-lessthan-or-equal-to-x-

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Question Number 147010 by gsk2684 last updated on 17/Jul/21

$$iftheradiusofacircletouching$$$$parabola{y}^2=4xat(4,4)andhaving$$$$directrixof{y}^2=4xasitsnormal$$$$isr,thenfind\left[r\right]?$$$$(where\left[x\right]denotegreatestinteger$$$$lessthanorequaltox)$$

Commented by gsk2684 last updated on 17/Jul/21

$$helpme$$

Answered by mr W last updated on 17/Jul/21

$$x=\frac{y^2}{4}$$$$directrixisx=-1$$$$\frac{dx}{dy}=\frac{y}{2}$$$$at(4,4):$$$$\frac{dx}{dy}=\frac{4}{2}=2$$$$normalis$$$$y=4-2(x-4)=12-2x$$$$centerofcircle:$$$$x=-1$$$$y=12-2(-1)=14$$$$radiusr:$$$$r^2={(-1-4)}^2+{(14-4)}^2=125$$$$r=\sqrt{125}>\sqrt{121}=11,<\sqrt{144}=12$$$$\Rightarrow \left[r\right]=11$$

Commented by mr W last updated on 17/Jul/21

Commented by gsk2684 last updated on 18/Jul/21

$$thankyou$$

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