The number of different hyperbolas represented by the equation $(^{m} C_{n})^{2} (x + y + 1)^{2} = 2 {(x - 3)^{2} + (y - 4)^{2}}$ with $1 \leq n < m \leq 5$ is ___

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Solution

$3 (^{m} C_{n})^{2} {(\frac{x + y + 1}{\sqrt{2}})}^{2} = 2 ((x - 3)^{2} + (y - 4)^{2})^{m} C_{x} > 1 m = 2^{2} C_{1} = 2 m = 3^{3} C_{1} m = 4^{4} C_{1},^{4} C_{2} m = 5^{5} C_{1},^{5} C_{2}$
$∴ 6$ hyperbolas

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The number of different hyperbolas represented by the equation (mCn)2(x+y+1)2=2{(x−3)2+(y−4)2} with 1≤n<m≤5 is ___

3(mCn)2(x+y+1√2)2=2((x−3)2+(y−4)2)mCx>1m=22C1=2m=33C1m=44C1,4C2m=55C1,5C2 ∴6 hyperbolas

The number of different hyperbolas represented by the equation $(^{m} C_{n})^{2} (x + y + 1)^{2} = 2 {(x - 3)^{2} + (y - 4)^{2}}$ with $1 \leq n < m \leq 5$ is ___

$3 (^{m} C_{n})^{2} {(\frac{x + y + 1}{\sqrt{2}})}^{2} = 2 ((x - 3)^{2} + (y - 4)^{2})^{m} C_{x} > 1 m = 2^{2} C_{1} = 2 m = 3^{3} C_{1} m = 4^{4} C_{1},^{4} C_{2} m = 5^{5} C_{1},^{5} C_{2}$
$∴ 6$ hyperbolas