$$\left( 2ah'-2a'h \right) xy+\left( ab'-a'b \right) y^2=a-a'\\\left( ah'-a'h \right) x^2+\left( bh'-b'h \right) y^2=-\left( h-h' \right) \\\left( ab'-a'b \right) x^2+\left( 2hb'-2bh' \right) xy=-\left( b-b' \right) $$
$$\left( 2ah'-2a'h \right) xy+\left( ab'-a'b \right) y^2=a-a'\\\left( ah'-a'h \right) x^2+\left( bh'-b'h \right) y^2=-\left( h-h' \right) \\\left( ab'-a'b \right) x^2+\left( 2hb'-2bh' \right) xy=-\left( b-b' \right) $$
$$A: \left( 1 \right) \,\, The\,\,first\,\,result\,\,is\,\,nothing\,\,but\\\,\,\left( a-a' \right) x^2+2\left( h-h' \right) xy+\left( b-b' \right) y^2=0\\\varDelta \,\,is\,\,rational\Rightarrow \left( h-h' \right) ^2-\left( a-a' \right) \left( b-b' \right) \,\,is\,\,a\,\,square\,\,of\,\,rational\,\,numbers\\\left( 2 \right) \,\,The\,\,2nd\,\,part\,\,is\,\,not\,\,that\,\,easy\\\exp\text{ress }x^2\,\,in\,\,two\,\,different\,\,ways, we\,\,have\,\,\frac{1-2hxy-by^2}{a}=\frac{1-2h'xy-b'y^2}{a'}\\\Rightarrow \left( 2ah'-2a'h \right) xy+\left( ab'-a'b \right) y^2=a-a' ①\\\text{similiraly, }\exp\text{ress }xy\,\,and\,\,y^2, we\,\,have\\\left( ah'-a'h \right) x^2+\left( bh'-b'h \right) y^2=-\left( h-h' \right) \,\, ②\\\left( ab'-a'b \right) x^2+\left( 2hb'-2bh' \right) xy=-\left( b-b' \right) \\\text{e}\lim\text{inate x , u}\sin\text{g ①②,}$$
$$CH\text{1,}Q\text{19 }\left( sourse:tripos,1899 \right) \\\begin{array}{l} \,\,\text{19. If all the values of }x\,\,\text{and }y\,\,\text{given by }\\ \qquad ax^2+2hxy+by^2=\text{1,\quad }a^'x^2+2h^'xy+b^'y^2=1\\ \,\,\text{(where }a,h,b,a^',h^',b^'\,\,\text{are rational) are rational, then }\\ \qquad \left( h-h^' \right) ^2-\left( a-a^' \right) \left( b-b^' \right) ,\quad \left( ab^'-a^'b \right) ^2+4\left( ah^'-a^'h \right) \left( bh^'-b^'h \right)\\ \,\,\text{are both squares of rational numbers. }\\\end{array}\\A: \left( 1 \right) \,\, The\,\,first\,\,result\,\,is\,\,nothing\,\,but\\\,\,\left( a-a' \right) x^2+2\left( h-h' \right) xy+\left( b-b' \right) y^2=0\\\varDelta \,\,is\,\,rational\Rightarrow \left( h-h' \right) ^2-\left( a-a' \right) \left( b-b' \right) \,\,is\,\,a\,\,square\,\,of\,\,rational\,\,numbers\\\left( 2 \right) \,\,The\,\,2nd\,\,part\,\,is\,\,not\,\,that\,\,easy\\\exp\text{ress }x^2\,\,in\,\,two\,\,different\,\,ways, we\,\,have\,\,\frac{1-2hxy-by^2}{a}=\frac{1-2h'xy-b'y^2}{a'}\\\Rightarrow \left( 2ah'-2a'h \right) xy+\left( ab'-a'b \right) y^2=a-a' ①\\\text{similiraly, }\exp\text{ress }xy\,\,and\,\,y^2, we\,\,have\\\left( ah'-a'h \right) x^2+\left( bh'-b'h \right) y^2=-\left( h-h' \right) \,\, ②\\\left( ab'-a'b \right) x^2+\left( 2hb'-2bh' \right) xy=-\left( b-b' \right) \\\text{e}\lim\text{inate x , u}\sin\text{g ①②,}\\\left[ \left( ab'-a'b \right) y^2-\left( \text{a}-\text{a'} \right) \right] ^2=\left( \text{2ah'}-\text{2a'h} \right) ^2\text{x}^2\text{y}^2\\=4\left( \text{ah'}-\text{a'h} \right) ^2\text{y}^2\left[ -\frac{\left( \text{bh'}-\text{b'h} \right) \text{y}^2+\left( \text{h}-\text{h'} \right)}{\text{ah'}-\text{a'h}} \right] \\=-4\left( \text{ah'}-\text{a'h} \right) \text{y}^2\left[ \left( \text{bh'}-\text{b'h} \right) \text{y}^2+\left( \text{h}-\text{h'} \right) \right] \\\text{rearrange we have}\\\text{y}^4\left[ \left( \text{ab'}-\text{a'b} \right) ^2+4\left( \text{ah'}-\text{a'h} \right) \left( \text{bh'}-\text{b'h} \right) \right] +\text{y}^2\left[ 4\left( \text{ah'}-\text{a'h} \right) \left( \text{h}-\text{h'} \right) -\left( ab'-a'b \right) \left( \text{a}-\text{a'} \right) \right] +\left( \text{a}-\text{a'} \right) ^2=0\\\text{let m}=\left( \text{ab'}-\text{a'b} \right) ^2+4\left( \text{ah'}-\text{a'h} \right) \left( \text{bh'}-\text{b'h} \right) ,\text{n}=4\left( \text{ah'}-\text{a'h} \right) \left( \text{h}-\text{h'} \right) -\left( ab'-a'b \right) \left( \text{a}-\text{a'} \right) ,\text{t}=\left( \text{a}-\text{a'} \right) \\\text{thus my}^4+\text{ny}^2+\text{t}^2=0\\\text{then y}_1^2\text{y}_2^2=\frac{\text{t}^2}{\text{m}}\Rightarrow \text{m}=\left( \text{y}_1\text{y}_2\text{t} \right) ^2\Rightarrow \text{m is a square of rational number}\\\text{so }\left( ab^'-a^'b \right) ^2+4\left( ah^'-a^'h \right) \left( bh^'-b^'h \right) \,\,\text{is a square of rational number}$$
\\\left[ \left( ab'-a'b \right) y^2-\left( \text{a}-\text{a'} \right) \right] ^2=\left( \text{2ah'}-\text{2a'h} \right) ^2\text{x}^2\text{y}^2\\=4\left( \text{ah'}-\text{a'h} \right) ^2\text{y}^2\left[ -\frac{\left( \text{bh'}-\text{b'h} \right) \text{y}^2+\left( \text{h}-\text{h'} \right)}{\text{ah'}-\text{a'h}} \right] \\=-4\left( \text{ah'}-\text{a'h} \right) \text{y}^2\left[ \left( \text{bh'}-\text{b'h} \right) \text{y}^2+\left( \text{h}-\text{h'} \right) \right] \\\text{rearrange we have}\\\text{y}^4\left[ \left( \text{ab'}-\text{a'b} \right) ^2+4\left( \text{ah'}-\text{a'h} \right) \left( \text{bh'}-\text{b'h} \right) \right] +\text{y}^2\left[ 4\left( \text{ah'}-\text{a'h} \right) \left( \text{h}-\text{h'} \right) -\left( ab'-a'b \right) \left( \text{a}-\text{a'} \right) \right] +\left( \text{a}-\text{a'} \right) ^2=0\\\text{let m}=\left( \text{ab'}-\text{a'b} \right) ^2+4\left( \text{ah'}-\text{a'h} \right) \left( \text{bh'}-\text{b'h} \right) ,\text{n}=4\left( \text{ah'}-\text{a'h} \right) \left( \text{h}-\text{h'} \right) -\left( ab'-a'b \right) \left( \text{a}-\text{a'} \right) ,\text{t}=\left( \text{a}-\text{a'} \right) \\\text{thus my}^4+\text{ny}^2+\text{t}^2=0\\\text{then y}_1^2\text{y}_2^2=\frac{\text{t}^2}{\text{m}}\Rightarrow \text{m}=\left( \text{y}_1\text{y}_2\text{t} \right) ^2\Rightarrow \text{m is a square of rational number}\\\text{so }\left( ab^'-a^'b \right) ^2+4\left( ah^'-a^'h \right) \left( bh^'-b^'h \right) \,\,\text{is a square of rational number}
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