Math  /  Geometry

QuestionQuestion 5 10 pts
The directrix of the parabola y=112x26y=\frac{1}{12} x^{2}-6 is y=ly=l. What is the value of ll ? \square

Studdy Solution

STEP 1

1. The equation of the parabola is given in the form y=ax2+bx+c y = ax^2 + bx + c .
2. The parabola opens upwards or downwards, which is determined by the sign of a a .
3. The directrix of a parabola is a horizontal line when the parabola is in the form y=ax2+bx+c y = ax^2 + bx + c .

STEP 2

1. Identify the standard form of the parabola and its parameters.
2. Recall the formula for the directrix of a parabola.
3. Substitute the parameters into the formula.
4. Solve for the directrix.

STEP 3

Identify the standard form of the parabola and its parameters:
The given equation is y=112x26 y = \frac{1}{12}x^2 - 6 .
This is in the form y=ax2+bx+c y = ax^2 + bx + c where: - a=112 a = \frac{1}{12} - b=0 b = 0 - c=6 c = -6

STEP 4

Recall the formula for the directrix of a parabola:
For a parabola in the form y=ax2+bx+c y = ax^2 + bx + c , the directrix is given by:
y=c14a y = c - \frac{1}{4a}

STEP 5

Substitute the parameters into the formula for the directrix:
y=614×112 y = -6 - \frac{1}{4 \times \frac{1}{12}}

STEP 6

Solve for the directrix:
First, calculate the denominator:
4×112=412=13 4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3}
Now, substitute back into the formula:
y=6113 y = -6 - \frac{1}{\frac{1}{3}} y=63 y = -6 - 3 y=9 y = -9
Therefore, the value of l l is:
9 \boxed{-9}

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