Question

quiz 12 (8.0, 8.1, 8.2)
due saturday by 11:59pm points 6 submitting an external tool
quiz 12 (8.0, 8.1, 8.2)
score: 2.25/5 answered: 3/5
question 4
write the equation of the hyperbola ( 25 x^{2}-9 y^{2}-100 x + 18 y-134 = 0 ) in standard form
( \frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}} = 1 ),
where:
h=
k=
a=
b=
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Explanation:

Step1: Group x and y terms

Group the \(x\)-terms and \(y\)-terms together:
\(25x^2 - 100x - 9y^2 + 18y = 134\)

Step2: Factor out coefficients

Factor out the coefficients of the squared terms from the \(x\)-group and \(y\)-group:
\(25(x^2 - 4x) - 9(y^2 - 2y) = 134\)

Step3: Complete the square for x

For \(x^2 - 4x\), take half of \(-4\) (which is \(-2\)), square it (\(4\)), and add inside the \(x\)-group. Multiply by 25 outside:
\(25(x^2 - 4x + 4) - 9(y^2 - 2y) = 134 + 25 \times 4\)

Step4: Complete the square for y

For \(y^2 - 2y\), take half of \(-2\) (which is \(-1\)), square it (\(1\)), and add inside the \(y\)-group. Multiply by \(-9\) outside:
\(25(x^2 - 4x + 4) - 9(y^2 - 2y + 1) = 134 + 100 - 9 \times 1\)

Step5: Simplify both sides

Simplify the left side as perfect squares and the right side:
\(25(x - 2)^2 - 9(y - 1)^2 = 225\)

Step6: Divide by 225

Divide both sides by 225 to get the standard form:
\(\frac{(x - 2)^2}{9} - \frac{(y - 1)^2}{25} = 1\)

Answer:

\(h = 2\), \(k = 1\), \(a = 3\), \(b = 5\)