let-A-3-4-and-B-is-a-variable-point-on-the-line-x-6-if-AB-lt-4-then-the-number-of-position-of-B-with-integral-coordinates-is-please-help-

Question Number 98598 by MWSuSon last updated on 15/Jun/20

let A=(3,4) and B is a variable point on the line ∣x∣=6. if AB^(−) <4, then the number of position of B with integral coordinates is? please help!

$$\boldsymbol{{let}}\:\boldsymbol{{A}}=\left(\mathrm{3},\mathrm{4}\right)\:\boldsymbol{{and}}\:\boldsymbol{{B}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{variable}}\:\boldsymbol{{point}} \\ $$$$\boldsymbol{{on}}\:\boldsymbol{{the}}\:\boldsymbol{{line}}\:\mid\boldsymbol{{x}}\mid=\mathrm{6}.\:\boldsymbol{{if}}\:\overline {\boldsymbol{{AB}}}<\mathrm{4},\:\boldsymbol{{then}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{number}}\:\boldsymbol{{of}}\:\boldsymbol{{position}}\:\boldsymbol{{of}}\:\boldsymbol{{B}}\:\boldsymbol{{with}}\:\boldsymbol{{integral}} \\ $$$$\boldsymbol{{coordinates}}\:\boldsymbol{{is}}? \\ $$$$\boldsymbol{{please}}\:\boldsymbol{{help}}! \\ $$

Commented by bobhans last updated on 15/Jun/20

∣x∣ = 6 ⇒ { ((x = −6 or)),((x= 6)) :} case (1) let B(−6,y) then (√((y−4)^2 +81)) < 4 (y−4)^2 +81 < 16 ⇒y = ∅ case(2) let B(6,y) then (√((y−4)^2 +9)) < 4 (y−4)^2 −7 <0 ⇒(y−4−(√7))(y−4+(√7)) < 0 4 − (√7) < y < 4+(√7) 1.354 < y < 6.64 so position of B with integral coordinates where y ∈{2,3,4,5,6}

$$\mid{x}\mid\:=\:\mathrm{6}\:\Rightarrow\begin{cases}{{x}\:=\:−\mathrm{6}\:{or}}\\{{x}=\:\mathrm{6}}\end{cases} \\ $$$$\mathrm{case}\:\left(\mathrm{1}\right)\:\mathrm{let}\:\mathrm{B}\left(−\mathrm{6},\mathrm{y}\right)\:\mathrm{then}\:\sqrt{\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} +\mathrm{81}}\:<\:\mathrm{4} \\ $$$$\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} +\mathrm{81}\:<\:\mathrm{16}\:\Rightarrow\mathrm{y}\:=\:\varnothing \\ $$$$\mathrm{case}\left(\mathrm{2}\right)\:\mathrm{let}\:\mathrm{B}\left(\mathrm{6},\mathrm{y}\right)\:\mathrm{then}\:\sqrt{\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} +\mathrm{9}}\:<\:\mathrm{4} \\ $$$$\left(\mathrm{y}−\mathrm{4}\right)^{\mathrm{2}} −\mathrm{7}\:<\mathrm{0}\:\Rightarrow\left(\mathrm{y}−\mathrm{4}−\sqrt{\mathrm{7}}\right)\left(\mathrm{y}−\mathrm{4}+\sqrt{\mathrm{7}}\right)\:<\:\mathrm{0} \\ $$$$\mathrm{4}\:−\:\sqrt{\mathrm{7}}\:<\:\mathrm{y}\:<\:\mathrm{4}+\sqrt{\mathrm{7}}\: \\ $$$$\mathrm{1}.\mathrm{354}\:<\:\mathrm{y}\:<\:\mathrm{6}.\mathrm{64} \\ $$$$\mathrm{so}\:\mathrm{position}\:\mathrm{of}\:\mathrm{B}\:\mathrm{with}\:\mathrm{integral}\:\mathrm{coordinates} \\ $$$$\mathrm{where}\:\mathrm{y}\:\in\left\{\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\right\} \\ $$

Commented by MWSuSon last updated on 15/Jun/20

Thanks a lot.

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