What is d^2=(3-20t)^2+(4-30t)^2 ?

The solution to d^2=(3-20t)^2+(4-30t)^2 is Hyperbola with (h,k)=(0, 9/65),a=sqrt(1/13),b= 1/130

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d^2=(3-20t)^2+(4-30t)^2

d^{2} = (3 - 20 t)^{2} + (4 - 30 t)^{2}

(h, k) = (0, \frac{9}{65}), a = \frac{1}{13}, b = \frac{1}{130}

Solution steps

d^{2} = (3 - 20 t)^{2} + (4 - 30 t)^{2}

Rewrite

d^{2} - (3 - 20 t)^{2} - (4 - 30 t)^{2} = 0

in the form of a standard hyperbola equation

\frac{( d - 0 ) ^{2}}{( \frac{1}{13} ) ^{2}} - \frac{( t - \frac{9}{65} ) ^{2}}{( \frac{1}{130} ) ^{2}} = 1

Therefore Hyperbola properties are:

(h, k) = (0, \frac{9}{65}), a = \frac{1}{13}, b = \frac{1}{130}

x^{2} - y^{2} - 4 x - 6 y - 14 = 0

x^{2} - y^{2} - 6 x + 16 y - 119 = 0

(x - 2)^{2} - \frac{( y - 1 ) ^{2}}{4} = 1

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

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

What is d^2=(3-20t)^2+(4-30t)^2 ?
The solution to d^2=(3-20t)^2+(4-30t)^2 is Hyperbola with (h,k)=(0, 9/65),a=sqrt(1/13),b= 1/130