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EasonSYCEasonSYC
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finish q7 and step iii 2019. the diagrams took soooo long but they look nice.
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‎tex/2019/3/7-diag1.tex‎

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\begin{tikzpicture}
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\draw[->] (-5.5, 0) -- (5.5, 0) node[right] {\(x\)};
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\draw[->] (0, -5.5) -- (0, 5.5) node[above] {\(y\)};
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\draw (-5, -5) -- (5, 5) node [above right] {\(y = x\)};
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\draw (-5, 5) -- (5, -5) node [below right] {\(y = -x\)};
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\draw (0, 0) circle (3);
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\node at (2, 3.2) {\(x^2 + y^2 = a^2\)};
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\node at (3, 0) [below right] {\(a\)};
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\node at (-3, 0) [below left] {\(-a\)};
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\node at (0, 3) [above right] {\(a\)};
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\node at (0, -3) [below right] {\(-a\)};
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\end{tikzpicture}

‎tex/2019/3/7-diag2.tex‎

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\begin{tikzpicture}[scale=1.5]
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\draw[->] (-0.5, 0) -- (5.5, 0) node[right] {\(x\)};
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\draw[->] (0, -0.5) -- (0, 5.5) node[above] {\(y\)};
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\draw[smooth, samples=200, domain=0:1] plot ({sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- ({sqrt(2)}, 1);
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\draw[smooth, samples=200, domain=0:1] plot ({sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- ({sqrt(2)}, 1);
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\draw[smooth, samples=200, domain=2:sqrt(5)] plot ({sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- (0, {sqrt(5)});
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\draw[smooth, samples=200, domain=2:5] plot ({sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x);
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\draw[dotted] ({sqrt(2)}, 0) -- ({sqrt(2)}, 2) -- (0, 2);
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\draw[dotted] ({sqrt(2)}, 1) -- (0, 1);
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\node at (0, 0) [below left] {\(O\)};
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\node at (2, 0) [below] {\(2\)};
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\node at ({sqrt(2)}, 0) [below] {\(\sqrt{2}\)};
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\node at (0, 1) [left] {\(1\)};
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\node at (0, 2) [left] {\(2\)};
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\node at (0, {sqrt(5)}) [left] {\(\sqrt{5}\)};
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\draw[dashed] (0, 0) -- (5, 5) node[right] {\(y = x\)};
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\draw[dashed] (0, 0) -- (5, {2*sqrt(5)}) node [right] {\(y = \frac{2}{\sqrt{5}}x\)};
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\end{tikzpicture}

‎tex/2019/3/7-diag3.tex‎

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\begin{tikzpicture}[scale=1.2]
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\draw[->] (-5.5, 0) -- (5.5, 0) node[right] {\(x\)};
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\draw[->] (0, -5.5) -- (0, 5.5) node[above] {\(y\)};
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% 1st Quardrant (x, y)
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\draw[smooth, samples=200, domain=0:1] plot ({sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- ({sqrt(2)}, 1);
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\draw[smooth, samples=200, domain=0:1] plot ({sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- ({sqrt(2)}, 1);
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\draw[smooth, samples=200, domain=2:sqrt(5)] plot ({sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- (0, {sqrt(5)});
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\draw[smooth, samples=200, domain=2:5] plot ({sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x);
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% 2nd Quardrant (-x, y)
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\draw[smooth, samples=200, domain=0:1] plot ({-sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- ({-sqrt(2)}, 1);
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\draw[smooth, samples=200, domain=0:1] plot ({-sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- ({-sqrt(2)}, 1);
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\draw[smooth, samples=200, domain=2:sqrt(5)] plot ({-sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x) -- (0, {sqrt(5)});
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\draw[smooth, samples=200, domain=2:5] plot ({-sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, \x);
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% 3rd Quardrant (-x, -y)
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\draw[smooth, samples=200, domain=0:1] plot ({-sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x) -- ({-sqrt(2)}, -1);
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\draw[smooth, samples=200, domain=0:1] plot ({-sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x) -- ({-sqrt(2)}, -1);
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\draw[smooth, samples=200, domain=2:sqrt(5)] plot ({-sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x) -- (0, {-sqrt(5)});
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\draw[smooth, samples=200, domain=2:5] plot ({-sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x);
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% 4th Quardrant (x, -y)
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\draw[smooth, samples=200, domain=0:1] plot ({sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x) -- ({sqrt(2)}, -1);
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\draw[smooth, samples=200, domain=0:1] plot ({sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x) -- ({sqrt(2)}, -1);
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\draw[smooth, samples=200, domain=2:sqrt(5)] plot ({sqrt(-sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x) -- (0, {-sqrt(5)});
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\draw[smooth, samples=200, domain=2:5] plot ({sqrt(sqrt(\x^4 - 5*\x^2 + 4) + 2)}, -\x);
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\draw[dotted] ({sqrt(2)}, 2) -- ({sqrt(2)}, -2) -- (-{sqrt(2)}, -2) -- (-{sqrt(2)}, 2) -- cycle;
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\draw[dotted] ({-sqrt(2)}, 1) -- ({sqrt(2)}, 1);
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\draw[dotted] ({-sqrt(2)}, -1) -- ({sqrt(2)}, -1);
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\node at (0, 0) [below left] {\(O\)};
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\filldraw (2, 0) circle (0.8pt) node [below right] {\(2\)};
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\filldraw ({sqrt(2)}, 0) circle (0.8pt) node [below left] {\(\sqrt{2}\)};
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\filldraw (-2, 0) circle (0.8pt) node [below left] {\(-2\)};
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\filldraw ({-sqrt(2)}, 0) circle (0.8pt) node [below right] {\(-\sqrt{2}\)};
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\filldraw (0, 1) circle (0.8pt) node [above left] {\(1\)};
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\filldraw (0, 2) circle (0.8pt) node [below left] {\(2\)};
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\filldraw (0, {sqrt(5)}) circle (0.8pt) node [above left] {\(\sqrt{5}\)};
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\filldraw (0, -1) circle (0.8pt) node [below left] {\(-1\)};
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\filldraw (0, -2) circle (0.8pt) node [above left] {\(-2\)};
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\filldraw (0, {-sqrt(5)}) circle (0.8pt) node [below left] {\(-\sqrt{5}\)};
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\draw[dashed] (-5, -5) -- (5, 5) node[right] {\(y = x\)};
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\draw[dashed] (-5, 5) -- (5, -5) node[right] {\(y = -x\)};
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\draw[dashed] (-5, {-2*sqrt(5)}) -- (5, {2*sqrt(5)}) node [right] {\(y = \frac{2}{\sqrt{5}}x\)};
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\draw[dashed] (-5, {2*sqrt(5)}) -- (5, {-2*sqrt(5)}) node [right] {\(y = -\frac{2}{\sqrt{5}}x\)};
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\end{tikzpicture}

‎tex/2019/3/7.tex‎

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\Question{\currfilebase}
2-
\WorkInProgress
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\begin{enumerate}
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\item When \(a = b\),
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\begin{align*}
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y^2 (y^2 - a^2) & = x^2 (x^2 - a^2) \\
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x^4 - y^4 - a^2 x^2 + a^2 y^2 & = 0 \\
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(x^2 + y^2 - a^2) (x^2 - y^2) & = 0 \\
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(x^2 + y^2 - a^2) (x + y) (x - y) & = 0,
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\end{align*}
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so the Devil's Curve in this case consists of the line \(x + y = 0\), the line \(x - y = 0\), and the circle \(x^2 + y^2 = a^2\).
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The curve is shown as follows.
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\begin{center}
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\input{\currfiledir 7-diag1}
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\end{center}
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\item When \(a = 2\) and \(b = \sqrt{5}\),
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\[
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y^2 (y^2 - 5) = x^2 (x^2 - 4).
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\]
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\begin{enumerate}
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\item Rearrangement gives us
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\[
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(x^2)^2 - 4x^2 - y^2 (y^2 - 5) = 0,
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\]
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and considering the discriminant, we have
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\[
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(-4)^2 + 4 y^2 (y^2 - 5) \geq 0,
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\]
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i.e.
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\[
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\left(y^2 - 1\right) \left(y^2 - 4\right) \geq 0.
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\]
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This gives \(y^2 \leq 1\) or \(y^2 \geq 4\), and in the case where \(y \geq 0\), this must give \(0 \leq y \leq 1\) or \(y \geq 2\), as desired.
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\item When the curve is very close to the origin, we must have \(x^4, y^4 \ll x^2, y^2\), and hence \(4x^2 \approx 5y^2\), which means \(y \approx \frac{2}{\sqrt{5}}x\).
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When the curve is very far from the origin, we must have \(x^4, y^4 \gg x^2, y^2\), and hence \(x^4 \approx y^4\), which means \(y \approx x\).
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\item Using implicit differentiation, we have
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\begin{align*}
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y^2 (y^2 - 5) & = x^2 (x^2 - 4) \\
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(4y^3 - 10y) \DiffFrac{y}{x} & = 4x^3 - 8x \\
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(2y^2 - 5)y \DiffFrac{y}{x} & = 2x(x^2 - 2).
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\end{align*}
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When \(\DiffFrac{y}{x} = 0\), the tangent to the curve is parallel to the \(x\)-axis, and hence
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\[
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2x(x^2 - 2) = 0,
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\]
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giving \(x = 0\) or \(x = \sqrt{2}\).
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For \(x = 0\), \(y^2 (y^2 - 5) = 0\), and therefore \(y = 0\) or \(y = \sqrt{5}\). The case where \(y = 0\) does not necessarily give that \(\DiffFrac{y}{x} = 0\), but the case where \(y = \sqrt{5}\) does.
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For \(x = \sqrt{2}\), \(y^2 (y^2 - 5) = -4\), \(y = 2\) or \(y = 1\). Both cases give \(\DiffFrac{y}{x} = 0\).
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So the tangent to the curve is parallel to the \(x\)-axis at points
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\[
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\left(0, \sqrt{5}\right), \left(\sqrt{2}, 1\right), \left(\sqrt{2}, 2\right).
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\]
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We must have
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\[
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(2y^2 - 5)y = 2x(x^2 - 2) \DiffFrac{x}{y},
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\]
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and when \(\DiffFrac{x}{y} = 0\), the tangent to the curve is parallel to the \(y\)-axis.
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This gives \((2y^2 - 5)y = 0\), and hence \(y = 0\) or \(y = \sqrt{\frac{5}{2}}\).
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For \(y = 0\), \(x = 0\) or \(x = 2\). The case \(x = 0\) does not necessarily give \(\DiffFrac{x}{y} = 0\), but the case where \(x = 2\) does.
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For \(y = \sqrt{\frac{5}{2}}\), \(x^2 (x^2 - 4) = - \frac{25}{4}\), and hence
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\[
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4x^4 - 16x^2 + 25 = 4 (x^2 - 2)^2 + 9 = 0,
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\]
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which is not possible.
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Hence, the tangent to the curve is parallel to the \(y\)-axis only at \((2, 0)\).
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\end{enumerate}
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Therefore, from the analysis in the previous parts, the curve looks as follows for \(x \geq 0\) and \(y \geq 0\):
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\begin{center}
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\input{\currfiledir 7-diag2}
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\end{center}
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\item All \(x\) terms in the curve is in \(x^2\), so the graph is symmetric in the \(y\)-axis since \(x^2 = (-x)^2\). Similarly, the graph is symmetric in the \(x\)-axis as well. Hence, the complete graph looks as follows.
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\begin{center}
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\input{\currfiledir 7-diag3}
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\end{center}
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\end{enumerate}

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