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यूजर का नामheureka
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 #3
avatar+18845 
0

Consider all the points in the plane that solve the equation x^2 + 2y^2 = 16.
Find the maximum value of the product xy on this graph.  
(This graph is an example of an "ellipse".)

 

Formula: Ellipse equation
\(\begin{array}{|rcll|} \hline \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} &=& 1 \\ (xy)_{\text{max}} &=& \dfrac{ab}{2}\\ \hline \end{array}\)

 

Proof:

\(\begin{array}{|rcll|} \hline \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} &=& 1 \quad & | \quad \cdot x^2 \\\\ \dfrac{x^4}{a^2}+\dfrac{x^2y^2}{b^2} &=& x^2 \quad & | \quad \mathbf{z=xy} \\\\ \dfrac{x^4}{a^2}+\dfrac{\mathbf{z}^2}{b^2} &=& x^2 \\\\ \dfrac{x^4}{a^2}+\dfrac{\mathbf{z}^2}{b^2} &=& x^2 \quad & | \quad -\dfrac{x^4}{a^2} \\\\ \dfrac{\mathbf{z}^2}{b^2} &=& x^2 -\dfrac{x^4}{a^2} \quad & | \ \text{Differentiate each term with respect to x}\\\\ \dfrac{\mathbf{2z\ dz}}{b^2} &=& 2x\ dx -\dfrac{4x^3\ dx}{a^2} \quad & | \quad :2 \\\\ \dfrac{\mathbf{ z\ dz}}{b^2} &=& x\ dx -\dfrac{2x^3\ dx}{a^2} \quad & | \quad :\ dx \\\\ \dfrac{\mathbf{ z}}{b^2}\cdot\frac{\mathbf{dz}}{dx}\ &=& x-\dfrac{2x^3}{a^2} \quad & | \quad \text{set } \frac{ \mathbf{dz}}{dx} = 0 \Rightarrow \mathbf{ z}_{\text{max}} =(xy)_{\text{max}} \\\\ x-\dfrac{2x^3}{a^2} &=& 0 \\\\ x-\dfrac{2x^3}{a^2} &=& 0 \\\\ x\left(1- \dfrac{2x^2}{a^2} \right) &=& 0 \\ \hline \end{array}\)

 

1. \(x = 0\) no solution \(\Rightarrow xy = 0 \) no maximum

2. x = ?

\(\begin{array}{rcll} 1- \dfrac{2x^2}{a^2} &=& 0 \\\\ \dfrac{2x^2}{a^2} &=& 1 \\\\ x^2 &=& \dfrac{a^2}{2} \quad & | \quad \pm\sqrt{} \\\\ x &=& \pm \dfrac{a}{\sqrt{2}} \quad & | \quad \cdot \frac{\sqrt{2}}{\sqrt{2}} \\\\ \mathbf{ x } &\mathbf{=}& \mathbf{ \pm a\cdot \dfrac{\sqrt{2}}{2} } \end{array} \)

 

y = ?

\(\begin{array}{rcll} \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2} &=& 1 \quad & | \quad \mathbf{ x } &\mathbf{=}& \mathbf{ \pm \dfrac{a}{\sqrt{2}} } \\\\ \dfrac{a^2}{2a^2}+\dfrac{y^2}{b^2} &=& 1 \\\\ \dfrac{1}{2}+\dfrac{y^2}{b^2} &=& 1 \quad & | \quad -\dfrac12 \\\\ \dfrac{y^2}{b^2} &=& 1 -\dfrac12 \\\\ \dfrac{y^2}{b^2} &=& \dfrac12 \\\\ y^2 &=& \dfrac{b^2}{2} \quad & | \quad \pm\sqrt{} \\\\ y &=& \pm \dfrac{b}{\sqrt{2}} \quad & | \quad \cdot \frac{\sqrt{2}}{\sqrt{2}} \\\\ \mathbf{ y } &\mathbf{=}& \mathbf{ \pm b\cdot \dfrac{\sqrt{2}}{2} } \end{array}\)

 

xy = ?

\(\begin{array}{|rcll|} \hline \mathbf{ z}_{\text{max}} =(xy)_{\text{max}} \\ (xy)_{\text{max}} &=& \mathbf{ \left(\pm a\cdot \dfrac{\sqrt{2}}{2} \right) } \cdot \mathbf{ \left( \pm b\cdot \dfrac{\sqrt{2}}{2} \right) } \quad & | \quad xy > 0! \\\\ &=& a\cdot \dfrac{\sqrt{2}}{2}\cdot b\cdot \dfrac{\sqrt{2}}{2} \\\\ &=& ab\cdot \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{2}}{2} \\\\ \mathbf{(xy)_{\text{max}}} &\mathbf{=}& \mathbf{ \dfrac{ab}{2} } \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline x^2 + 2y^2 &=& 16 \quad & | \quad : 16 \\\\ \dfrac{x^2}{16} + \dfrac{2y^2}{16} &=& 1 \\\\ \dfrac{x^2}{16} + \dfrac{ y^2}{8} &=& 1 \\\\ \dfrac{x^2}{\mathbf{4}^2} + \dfrac{ y^2}{(\mathbf{\sqrt{8}})^2} &=& 1 \\\\ \text{so } a = 4 \text{ and } b = \sqrt{8} \\\\ \mathbf{(xy)_{\text{max}}} &\mathbf{=}& \mathbf{ \dfrac{ab}{2} } \\\\ &=& \dfrac{4 \sqrt{8}}{2} \\\\ &=& 2 \sqrt{8} \\ &=& 2 \sqrt{2\cdot4 } \\ &=& 2\cdot2 \sqrt{2} \\ \mathbf{(xy)_{\text{max}}} &\mathbf{=}& \mathbf{4 \sqrt{2}} \\ \hline \end{array} \)

 

\(\begin{array}{rcll} Point_1 &=& ( a\cdot \dfrac{\sqrt{2}}{2},~ b\cdot \dfrac{\sqrt{2}}{2} ) \\ &=& ( 4\cdot \dfrac{\sqrt{2}}{2},~ \sqrt{8}\cdot \dfrac{\sqrt{2}}{2} ) \\ &=& ( 2\cdot \sqrt{2} ,~ \dfrac{\sqrt{16}}{2} ) \\ &=& ( 2\cdot \sqrt{2} ,~ \dfrac{4}{2} ) \\ &=& ( 2\cdot \sqrt{2} ,~ 2 ) \\ \end{array} \begin{array}{rcll} Point_2 &=& ( -a\cdot \dfrac{\sqrt{2}}{2},~ -b\cdot \dfrac{\sqrt{2}}{2} ) \\ &=& ( -4\cdot \dfrac{\sqrt{2}}{2}, -\sqrt{8}\cdot \dfrac{\sqrt{2}}{2} ) \\ &=& ( -2\cdot \sqrt{2} ,~ -\dfrac{\sqrt{16}}{2} ) \\ &=& ( -2\cdot \sqrt{2} ,~ -\dfrac{4}{2} ) \\ &=& ( -2\cdot \sqrt{2} ,~ -2 ) \\ \end{array} \)

 

laugh

heureka 36 mins ago
 #2
avatar+18845 
+1

A palindrome is a positive integer which reads the same forward and backwards, like  12321 or 8448.
How many 4-digit palindromes are divisible by 3?

 

\(\begin{array}{|r|r|r|r|} \hline & n & \text{palindrome } a(n) & \text{divisible by } 3 \\ \hline & 110& 10\ 01 \\ & 111& 11\ 11 \\ 1. & 112& 12\ 21 & \checkmark \\ & 113& 13\ 31 \\ & 114& 14\ 41 \\ 2. & 115& 15\ 51 &\checkmark \\ & 116& 16\ 61 \\ & 117& 17\ 71 \\ 3. & 118& 18\ 81 &\checkmark \\ & 119& 19\ 91 \\ \hline & 120& 20\ 02 \\ 4. & 121& 21\ 12 &\checkmark \\ & 122& 22\ 22 \\ & 123& 23\ 32 \\ 5. & 124& 24\ 42 &\checkmark \\ & 125& 25\ 52 \\ & 126& 26\ 62 \\ 6. & 127& 27\ 72 &\checkmark \\ & 128& 28\ 82 \\ & 129& 29\ 92 \\ \hline 7. & 130& 30\ 03 &\checkmark \\ & 131& 31\ 13 \\ & 132& 32\ 23 \\ 8. & 133& 33\ 33 &\checkmark \\ & 134& 34\ 43 \\ & 135& 35\ 53 \\ 9. & 136& 36\ 63 &\checkmark \\ & 137 & 37\ 73 \\ & 138& 38\ 83 \\ 10. & 139& 39\ 93 &\checkmark \\ \hline & 140& 40\ 04 \\ & 141& 41\ 14 \\ 11. & 142& 42\ 24 &\checkmark \\ & 143& 43\ 34 \\ & 144& 44\ 44 \\ 12. & 145& 45\ 54 &\checkmark \\ & 146& 46\ 64 \\ & 147& 47\ 74 \\ 13. & 148& 48\ 84 &\checkmark \\ & 149& 49\ 94 \\ \hline & 150& 50\ 05 \\ 14. & 151& 51\ 15 &\checkmark \\ & 152& 52\ 25 \\ & 153& 53\ 35 \\ 15. & 154& 54\ 45 &\checkmark \\ & 155& 55\ 55 \\ & 156& 56\ 65 \\ 16. & 157& 57\ 75 &\checkmark \\ & 158& 58\ 85 \\ & 159& 59\ 95 \\ \hline 17. & 160& 60\ 06 &\checkmark \\ & 161& 61\ 16 \\ & 162& 62\ 26 \\ 18. & 163& 63\ 36 &\checkmark \\ & 164& 64\ 46 \\ & 165& 65\ 56 \\ 19. & 166& 66\ 66 &\checkmark \\ & 167& 67\ 76 \\ & 168& 68\ 86 \\ 20. & 169& 69\ 96 &\checkmark \\ \hline & 170& 70\ 07 \\ & 171& 71\ 17 \\ 21. & 172& 72\ 27 &\checkmark \\ & 173& 73\ 37 \\ & 174& 74\ 47 \\ 22. & 175& 75\ 57 &\checkmark \\ & 176& 76\ 67 \\ & 177& 77\ 77 \\ 23. & 178& 78\ 87 &\checkmark \\ & 179& 79\ 97 \\ \hline & 180& 80\ 08 \\ 24. & 181& 81\ 18 &\checkmark \\ & 182& 82\ 28 \\ & 183& 83\ 38 \\ 25. & 184& 84\ 48 &\checkmark \\ & 185& 85\ 58 \\ & 186& 86\ 68 \\ 26. & 187& 87\ 78 &\checkmark \\ & 188& 88\ 88 \\ & 189& 89\ 98 \\ \hline 27. & 190& 90\ 09 &\checkmark \\ & 191& 91\ 19 \\ & 192& 92\ 29 \\ 28. & 193& 93\ 39 &\checkmark \\ & 194& 94\ 49 \\ & 195& 95\ 59 \\ 29. & 196& 96\ 69 &\checkmark \\ & 197& 97\ 79 \\ & 198& 98\ 89 \\ 30. & 199& 99\ 99 &\checkmark \\ \hline \end{array}\)

 

30  4-digit palindromes are divisible by 3.

 

laugh

heureka 2018 Jan 23 13:13
 #1
avatar+18845 
+2

I don't get eccentricity and its relation to hyperbolas...in a question where I have to form an equation for a hyperbola with a center at (-3,1), one focus at (2,1), and eccentricity is 5/4, how would I solve this? 

 

\(center : ~(h=-3,~ k=1)\)

\(focus: ~( x_f = 2,~ y_f=1)\)

\(eccentricity: ~e = \frac{5}{4}\)

 

Because \(k = y_f =1\) the transverse axis is horizontal.

 

1.  c = ?

\(\begin{array}{|rcll|} \hline focus: ( h+c,~k ) &=& (2,1)\\ h+c &=& 2 \\ -3+c &=& 2 \\ c&=& 2+3\\ \mathbf{c} &\mathbf{=}& \mathbf{5} \\ \hline \end{array}\)

 

(2.)

\(\begin{array}{|rcll|} \hline focus_2: ( h-c,~k ) &=& (x_f,y_f=k)\\ h-c &=& x_f \\ -3-5 &=& x_f \\ x_f &=& -8\\ y_f &=& 1 \\ \mathbf{focus_2} &\mathbf{=}& \mathbf{(-8,1)} \\ \hline \end{array} \)

 

3. a = ?

\(\begin{array}{|rcll|} \hline \mathbf{e} &\mathbf{=}& \mathbf{\frac{c}{a}} \\ \frac{5}{4} &=& \frac{5}{a} \\ \frac{4}{5} &=& \frac{a}{5} \\ a &=& \frac{4}{5}\cdot 5 \\ \mathbf{a} &\mathbf{=}& \mathbf{4} \\ \hline \end{array}\)

 

4. b = ?

\(\begin{array}{|rcll|} \hline \mathbf{c^2} &\mathbf{=}& \mathbf{a^2+b^2} \\ b^2 &=&c^2-a^2 \\ b^2 &=&5^2-4^2 \\ b^2 &=&25-16 \\ b^2 &=& 9 \\ \mathbf{b} &\mathbf{=}& \mathbf{3} \\ \hline \end{array}\)

 

5.  Equation:

\(\begin{array}{|rcll|} \hline \mathbf{\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}} &\mathbf{=}& \mathbf{1} \\ \frac{(x-(-3))^2}{4^2}-\frac{(y-1)^2}{3^2} &=& 1 \\ \mathbf{\frac{(x+3)^2}{4^2}-\frac{(y-1)^2}{3^2}} &\mathbf{=}& \mathbf{1} \\ \hline \end{array} \)

 

6. The graph:

 

 

laugh

heureka 2018 Jan 22
 #6
avatar+18845 
0

Compute the sum    2/(1*2*3) + 2/(2*3*4) + 2/(3*4*5)+...

 

see link: http://web2.0calc.com/questions/algebra_47009

 

In General:

\(\begin{array}{lcll} s = \dfrac{1}{1 \cdot 2 } + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4 } + \dfrac{1}{4 \cdot 5 } + \cdots \ + \dfrac{1}{n \cdot (n+1)} + \cdots =\ \frac{1}{1!0!}\cdot \frac{1}{1} = 1 \\\\ s =\mathbf{ \dfrac{1}{1 \cdot 2 \cdot 3} + \dfrac{1}{2 \cdot 3 \cdot 4} + \dfrac{1}{3 \cdot 4 \cdot 5} + \dfrac{1}{4 \cdot 5 \cdot 6} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2)} + \cdots =\ \frac{1}{2!0!}\cdot \frac{1}{2} } = \frac{1}{4} \\\\ s = \dfrac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \dfrac{1}{2 \cdot 3 \cdot 4 \cdot 5} + \dfrac{1}{3 \cdot 4 \cdot 5 \cdot 6} + \dfrac{1}{4 \cdot 5 \cdot 6 \cdot 7} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2) \cdot (n+3)} + \cdots =\ \frac{1}{3!0!}\cdot \frac{1}{3} = \frac{1}{18} \\\\ s = \dfrac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \dfrac{1}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6} + \dfrac{1}{3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \dfrac{1}{4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2) \cdot (n+3 \cdot (n+4) } + \cdots =\ \frac{1}{4!0!}\cdot \frac{1}{4} = \frac{1}{96} \\\\ \ldots \\ s = \dfrac{1}{1 \cdot 2 \cdot 3 \cdots } + \dfrac{1}{2 \cdot 3 \cdot 4 \cdots } + \dfrac{1}{3 \cdot 4 \cdot 5 \cdots} + \cdots \ + \dfrac{1}{n \cdot (n+1) \cdot (n+2)\cdot~\cdots ~\cdot (n+m) } + \cdots =\ \frac{1}{m!0!}\cdot \frac{1}{m} = \frac{1}{m\cdot m!} = \frac{1}{(m+1)!-m!} \\\\ \end{array} \\ \)

 

 

 

 

laugh

heureka 2018 Jan 19
 #3
avatar+18845 
+1

A circle of radius 5 with its center at $(0,0)$ is drawn on a Cartesian coordinate system.

How many lattice points (points with integer coordinates) lie within or on this circle?

 

A Calculation of the Number of Lattice Points within or on the circle:

 

Let \( \lfloor x \rfloor \) be the largest integer equal to or less than x.

 

Example:
\(\lfloor 3.53553390593 \rfloor = 3\)
\(\lfloor -3.53553390593 \rfloor = -4\)

 

 

 

Noted by Gauss:

Let r  radius of the circle = 5

Let \(x = r^2\)

 

\(\begin{array}{|rcll|} \hline A_2(x) &=& 1 + 4\lfloor \sqrt{x} \rfloor + 4 \lfloor \sqrt{\frac{x}{2}} \rfloor ^2 + 8 \sum \limits_{y_1= \lfloor \sqrt{\frac{x}{2}} \rfloor + 1 }^{\lfloor \sqrt{x} \rfloor} \lfloor \sqrt{x-y_1^2} \rfloor \qquad & | \quad x = r^2 = 5^2 \\\\ &=& 1 + 4\lfloor \sqrt{5^2} \rfloor + 4 \lfloor \sqrt{\frac{5^2}{2}} \rfloor ^2 + 8 \sum \limits_{y_1= \lfloor \sqrt{\frac{5^2}{2}} \rfloor + 1 }^{\lfloor \sqrt{5^2} \rfloor} \lfloor \sqrt{5^2-y_1^2} \rfloor \\\\ &=& 1 + 4 \cdot 5 + 4 \cdot 3 ^2 + 8 \sum \limits_{y_1= 3 + 1 }^{5} \lfloor \sqrt{5^2-y_1^2} \rfloor \\\\ &=& 1 + 4 \cdot 5 + 4 \cdot 3 ^2 + 8 \sum \limits_{y_1= 4 }^{5} \lfloor \sqrt{5^2-y_1^2} \rfloor \\\\ &=& 1 + 4 \cdot 5 + 4 \cdot 3 ^2 + 8 \cdot \left( \lfloor \sqrt{5^2-4^2} \rfloor +\lfloor \sqrt{5^2-5^2} \rfloor \right) \\\\ &=& 1 + 4 \cdot 5 + 4 \cdot 3 ^2 + 8 \cdot \left( 3 + 0 \right) \\\\ &=& 1 + 4 \cdot 5 + 4 \cdot 3 ^2 + 24 \\\\ &=& 1 + 20 + 36 + 24 \\ &\mathbf{=} & \mathbf{81} \\ \hline \end{array}\)

 

81 lattice points (points with integer coordinates) lie within or on this circle with radius 5.

 

Example:
\(r = 0 \ldots 20\)

 

Number of lattice points in circle:

\(\begin{array}{|r|r|r|} \hline r & \text{lattice points in circle} & \text{lattice points in sphere } \\ \hline 0 & 1 & 1 \\ 1 & 5 & 7 \\ 2 & 13 & 33 \\ 3 & 29 & 123 \\ 4 & 49 & 257 \\ {\color{red}5} & {\color{red}81} & 515 \\ 6 & 113 & 925 \\ 7 & 149 & 1419 \\ 8 & 197 & 2109 \\ 9 & 253 & 3071 \\ 10 & 317 & 4169 \\ 11 & 377 & 5575 \\ 12 & 441 & 7153 \\ 13 & 529 & 9171 \\ 14 & 613 & 11513 \\ 15 & 709 & 14147 \\ 16 & 797 & 17077 \\ 17 & 901 & 20479 \\ 18 & 1009 & 24405 \\ 19 & 1129 & 28671 \\ 20 & 1257 & 33401 \\ \hline \end{array} \)

 

laugh

heureka 2018 Jan 15