If \( x^2 = -16y \) is an equation of a parabola, then:
(A) Directrix is \( y = 4 \)
(B) Directrix is \( x = 4 \)
(C) Co-ordinates of focus are \( (0, -4) \)
(D) Co-ordinates of focus are \( (-4, 0) \)
(E) Length of latus rectum is 16
Step 1: Standard form of the parabola.
The equation \( x^2 = -16y \) represents a parabola opening downwards. Its standard form is:\[x^2 = -4ay\]Comparing this with the given equation, we find \( 4a = 16 \), which implies \( a = 4 \).
Step 2: Determine the focus and directrix.
- The focus is located at \( (0, -a) \), which is \( (0, -4) \). - The directrix is the line \( y = a \), which is \( y = 4 \).
Step 3: Calculate the length of the latus rectum.
The length of the latus rectum for any parabola is \( 4a \). In this case, \( 4a = 16 \).Therefore, the correct answer is 3. (A), (C), and (E) exclusively.
Two parabolas have the same focus $(4, 3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
Let \( y^2 = 12x \) be the parabola and \( S \) its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ) = \frac{147}{4} \). Let \( C \) be the circle described by taking \( PQ \) as a diameter. If the equation of the circle \( C \) is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then \( \beta - \alpha \) is equal to: