Find the focus and directrix of the following parabola: $(y-6)^2=12(x-2)$ Focus: ([ ? ], $\square$ ) Directrix: $x=$ $\square$

The given equation is ( y − 6 ) 2 = 12 ( x − 2 ) , which represents a parabola.
By comparing with the standard form ( y − k ) 2 = 4 p ( x − h ) , we identify h = 2 , k = 6 , and 4 p = 12 .
Solving for p , we find p = 3 .
The focus is ( h + p , k ) = ( 5 , 6 ) and the directrix is x = h − p = − 1 . Therefore, the focus is ( 5 , 6 ) ​ and the directrix is x = − 1 ​ .

Explanation

Problem Analysis We are given the equation of a parabola: ( y − 6 ) 2 = 12 ( x − 2 ) . Our goal is to find the focus and directrix of this parabola.

Recall the Standard Form of a Parabola The standard form of a parabola that opens to the right is given by ( y − k ) 2 = 4 p ( x − h ) , where ( h , k ) is the vertex of the parabola, and p is the distance from the vertex to the focus and from the vertex to the directrix. The focus is located at ( h + p , k ) , and the directrix is the vertical line x = h − p .

Identify h, k, and 4p Comparing the given equation ( y − 6 ) 2 = 12 ( x − 2 ) with the standard form ( y − k ) 2 = 4 p ( x − h ) , we can identify the values of h , k , and p . We have:


h = 2 k = 6 4 p = 12

Solve for p From 4 p = 12 , we can solve for p by dividing both sides by 4:

p = 4 12 ​ = 3

Calculate the Focus and Directrix Now that we have the values of h , k , and p , we can find the focus and directrix.

The focus is at ( h + p , k ) = ( 2 + 3 , 6 ) = ( 5 , 6 ) .
The directrix is the line x = h − p = 2 − 3 = − 1 .

State the Final Answer Therefore, the focus of the parabola is ( 5 , 6 ) , and the directrix is x = − 1 .

Focus: ( 5 , 6 ) Directrix: x = − 1
Examples
Parabolas are commonly used in the design of satellite dishes and reflecting telescopes. The reflective property of a parabola ensures that incoming parallel rays (like signals from a satellite) are focused at a single point (the focus), where the receiver is placed. Similarly, headlights in cars use a parabolic reflector to project light in a parallel beam, providing focused illumination on the road.

Answered by GinnyAnswer | 2025-07-05

The focus of the parabola is at the point (5, 6), and the directrix is the line x = -1.
;

Answered by Anonymous | 2025-07-06