Question

Find the coordinates of the focus, axis of the parabola, the equation of the directrix, and the length of the latus rectum, if \( y^2 = 12x \).

Hint:

For a parabola of the form \( y^2 = 4ax \), the focus is at \( (a, 0) \), the axis is the x-axis, and the directrix is the line \( x = -a \).

Answer 1

Step 1: Recognize the standard form of the equation.
The equation \( y^2 = 12x \) is in the standard form of a parabola that opens to the right, which is: \[ y^2 = 4ax \] where \( a \) is the distance from the vertex to the focus.
Step 2: Compare with the standard equation.}
Comparing \( y^2 = 12x \) with \( y^2 = 4ax \), we get \( 4a = 12 \), so \( a = 3 \).
Step 3: Find the coordinates of the focus.}
The focus of a parabola \( y^2 = 4ax \) is at \( (a, 0) \). Therefore, the coordinates of the focus are \( (3, 0) \).
Step 4: Find the equation of the directrix.}
The directrix of the parabola \( y^2 = 4ax \) is given by \( x = -a \). Therefore, the equation of the directrix is: \[ x = -3 \]
Step 5: Find the axis of the parabola.}
The axis of the parabola is the line that passes through the focus and the vertex. In this case, the axis is the x-axis, or \( y = 0 \).
Step 6: Find the length of the latus rectum.}
The length of the latus rectum for a parabola \( y^2 = 4ax \) is given by \( 4a \). Therefore, the length of the latus rectum is: \[ 4a = 4 \times 3 = 12 \] So, the answers are: - Focus: \( (3, 0) \) - Axis: \( y = 0 \) - Directrix: \( x = -3 \) - Length of the latus rectum: 12