(74x3+53y8)(74x3-53y8)=4916x6-259y16

解答

(74 x 3 + 53 y 8) (74 x 3 - 53 y 8) = 4916 x 6 - 259 y 16

(h, k) = (\frac{1229}{4107}, \frac{259}{22472}), a = 424 \frac{1510441}{3032362511424} - \frac{49}{18179668224}, b = 222 \frac{1510441}{3032362511424} - \frac{49}{18179668224}

求解步骤

(74 x \cdot 3 + 53 y \cdot 8) (74 x \cdot 3 - 53 y \cdot 8) = 4916 x \cdot 6 - 259 y \cdot 16

将

(74 x 3 + 53 y 8) (74 x 3 - 53 y 8) - 4916 x 6 + 259 y 16 = 0

改写为标准双曲线方程形式

\frac{( x - \frac{1229}{4107} ) ^{2}}{( 424 \frac{1510441}{3032362511424} - \frac{49}{18179668224} ) ^{2}} - \frac{( y - \frac{259}{22472} ) ^{2}}{( 222 \frac{1510441}{3032362511424} - \frac{49}{18179668224} ) ^{2}} = 1

因此双曲线的性质为:

(h, k) = (\frac{1229}{4107}, \frac{259}{22472}), a = 424 \frac{1510441}{3032362511424} - \frac{49}{18179668224}, b = 222 \frac{1510441}{3032362511424} - \frac{49}{18179668224}

- 12 x^{2} + 2 y^{2} + 12 x - 4 y + 4 = 0

a^{2} - b^{2} = 9

\frac{( x + 4 ) ^{2}}{9} - \frac{( y + 3 ) ^{2}}{16} = 1

x^{2} - 4 y^{2} - 8 = 0

x^{2} - y^{2} - 4 x + 8 y - 21 = 0