Question

Write the equation in the standard form of a conic.
−(x)^2+4x+3y^2−18y+8=0

Answers

GPT-4o mini
To convert the given equation \(-x^2 + 4x + 3y^2 - 18y + 8 = 0\) into standard form, we will follow these steps: rearranging the equation, completing the square for both \(x\) and \(y\), and then rewriting it in standard form.

1. **Rearrange the equation:**
\[
-x^2 + 4x + 3y^2 - 18y + 8 = 0
\]
Move \(8\) to the right side:
\[
-x^2 + 4x + 3y^2 - 18y = -8
\]

2. **Factor out the coefficients of squares:**
We need to factor out \(-1\) from the \(x\) terms:
\[
-(x^2 - 4x) + 3y^2 - 18y = -8
\]

3. **Complete the square for \(x\):**
Take \(x^2 - 4x\):
\[
x^2 - 4x = (x - 2)^2 - 4
\]
Then we have:
\[
-( (x - 2)^2 - 4 ) + 3y^2 - 18y = -8
\]
Simplifying gives:
\[
-(x - 2)^2 + 4 + 3y^2 - 18y = -8
\]
Combine the constant terms:
\[
- (x - 2)^2 + 3y^2 - 18y + 4 = -8
\]
Move 4 to the right side:
\[
- (x - 2)^2 + 3y^2 - 18y = -12
\]

4. **Complete the square for \(y\):**
Take \(3y^2 - 18y\):
\[
3(y^2 - 6y) = 3((y - 3)^2 - 9) = 3(y - 3)^2 - 27
\]
Substituting back gives:
\[
- (x - 2)^2 + (3(y - 3)^2 - 27) = -12
\]
Simplifying leads to:
\[
- (x - 2)^2 + 3(y - 3)^2 - 27 = -12
\]
Move \(-27\) to the right side:
\[
- (x - 2)^2 + 3(y - 3)^2 = 15
\]

5. **Multiply through by \(-1\) to rewrite:**
\[
(x - 2)^2 - 3(y - 3)^2 = -15
\]
This can be rearranged to:
\[
(x - 2)^2 = 3(y - 3)^2 - 15
\]
To get it into the standard form for a hyperbola:
\[
\frac{(x - 2)^2}{15} - \frac{(y - 3)^2}{5} = 1
\]

The final form is:
\[
\frac{(x - 2)^2}{15} - \frac{(y - 3)^2}{5} = 1
\]

This represents a hyperbola in standard form.

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