Classify the conic section defined by the equation \(4x^2 - 9y^2 + 8x + 36y - 68 = 0\) and write its standard equation.
Group the \(x\) and \(y\) terms.
\( (4x^2 + 8x) - (9y^2 - 36y) = 68 \)
Complete the square for both \(x\) and \(y\).
\( 4(x^2 + 2x) - 9(y^2 - 4y) = 68 \)
\( 4(x^2 + 2x + 1) - 9(y^2 - 4y + 4) = 68 + 4(1) - 9(4) \)
\( 4(x + 1)^2 - 9(y - 2)^2 = 68 + 4 - 36 \)
\( 4(x + 1)^2 - 9(y - 2)^2 = 36 \)
Divide by 36 to get the standard form.
\( \frac{(x + 1)^2}{9} - \frac{(y - 2)^2}{4} = 1 \)
\(\boxed{\frac{(x + 1)^2}{9} - \frac{(y - 2)^2}{4} = 1}\)
Classify the conic section defined by the equation \(4x^2 + 16y^2 - 8x + 64y = 28\) and write its standard equation.
Group the \(x\) and \(y\) terms.
\( (4x^2 - 8x) + (16y^2 + 64y) = 28 \)
Complete the square for both \(x\) and \(y\).
\( 4(x^2 - 2x) + 16(y^2 + 4y) = 28 \)
\( 4(x^2 - 2x + 1) + 16(y^2 + 4y + 4) = 28 + 4(1) + 16(4) \)
\( 4(x - 1)^2 + 16(y + 2)^2 = 28 + 4 + 64 \)
\( 4(x - 1)^2 + 16(y + 2)^2 = 96 \)
Divide by 96 to get the standard form.
\( \frac{(x - 1)^2}{24} + \frac{(y + 2)^2}{6} = 1 \)
\(\boxed{\frac{(x - 1)^2}{24} + \frac{(y + 2)^2}{6} = 1}\)
Classify the conic section defined by the equation \(y^2 - 3y - 4x + 3 = 0\) and write its standard equation.
Rearrange the equation.
\( y^2 - 3y = 4x - 3 \)
Complete the square for \(y\).
\( y^2 - 3y + \left(\frac{3}{2}\right)^2 = 4x - 3 + \left(\frac{3}{2}\right)^2 \)
\( \left(y - \frac{3}{2}\right)^2 = 4x - 3 + \frac{9}{4} \)
\( \left(y - \frac{3}{2}\right)^2 = 4x - \frac{12}{4} + \frac{9}{4} \)
\( \left(y - \frac{3}{2}\right)^2 = 4x - \frac{3}{4} \)
Write the standard equation.
\( \left(y - \frac{3}{2}\right)^2 = 4\left(x - \frac{3}{16}\right) \)
\(\boxed{\left(y - \frac{3}{2}\right)^2 = 4\left(x - \frac{3}{16}\right)}\)
Classify the conic section defined by the equation \(3x^2 + 5y^2 + 18x - 20y + 2 = 0\) and write its standard equation.
Group the \(x\) and \(y\) terms.
\( (3x^2 + 18x) + (5y^2 - 20y) = -2 \)
Complete the square for both \(x\) and \(y\).
\( 3(x^2 + 6x) + 5(y^2 - 4y) = -2 \)
\( 3(x^2 + 6x + 9) + 5(y^2 - 4y + 4) = -2 + 3(9) + 5(4) \)
\( 3(x + 3)^2 + 5(y - 2)^2 = -2 + 27 + 20 \)
\( 3(x + 3)^2 + 5(y - 2)^2 = 45 \)
Divide by 45 to get the standard form.
\( \frac{(x + 3)^2}{15} + \frac{(y - 2)^2}{9} = 1 \)
\(\boxed{\frac{(x + 3)^2}{15} + \frac{(y - 2)^2}{9} = 1}\)
Classify the conic section defined by the equation \(4y^2 - 36x^2 - 72x + 8y = 176\) and write its standard equation.
Group the \(x\) and \(y\) terms.
\( (4y^2 + 8y) - (36x^2 + 72x) = 176 \)
Complete the square for both \(x\) and \(y\).
\( 4(y^2 + 2y) - 36(x^2 + 2x) = 176 \)
\( 4(y^2 + 2y + 1) - 36(x^2 + 2x + 1) = 176 + 4(1) - 36(1) \)
\( 4(y + 1)^2 - 36(x + 1)^2 = 176 + 4 - 36 \)
\( 4(y + 1)^2 - 36(x + 1)^2 = 144 \)
Divide by 144 to get the standard form.
\( \frac{(y + 1)^2}{36} - \frac{(x + 1)^2}{4} = 1 \)
\(\boxed{\frac{(y + 1)^2}{36} - \frac{(x + 1)^2}{4} = 1}\)
\(\boxed{\frac{(x + 1)^2}{9} - \frac{(y - 2)^2}{4} = 1}\)
\(\boxed{\frac{(x - 1)^2}{24} + \frac{(y + 2)^2}{6} = 1}\)
\(\boxed{\left(y - \frac{3}{2}\right)^2 = 4\left(x - \frac{3}{16}\right)}\)
\(\boxed{\frac{(x + 3)^2}{15} + \frac{(y - 2)^2}{9} = 1}\)
\(\boxed{\frac{(y + 1)^2}{36} - \frac{(x + 1)^2}{4} = 1}\)
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