MHB Find Parabola Given Focus & Directrix - Help Needed

  • Thread starter Thread starter MarkFL
  • Start date Start date
  • Tags Tags
    Focus Parabola
AI Thread Summary
To derive the equation of a parabola with a focus at (-5, -5) and a directrix of y = 7, the relationship between the focus and directrix is used. The equation is established using the distance formula, leading to the quadratic function y = -\frac{x^2 + 10x + 1}{24}. The vertex of the parabola is located at (-5, 1), with the directed distance from the vertex to the focus being -6. Consequently, the final equation of the parabola is (x + 5)^2 = -24(y - 1). This provides a complete representation of the parabola based on the given parameters.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Find the formula of this parabola?


Derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7.

So... I've tried this one over and over but can't seem to get the right answer. Help anyone?

I have posted a link there to this topic so the OP can see my work.
 
Mathematics news on Phys.org
Hello Abs,

A parabola is defined as the locus of all points $(x,y)$ equidistant from a point (the focus) and a line (the directrix). Using the square of the distance formula, we may write:

$$(x+5)^2+(y+5)^2=(y-7)^2$$

$$x^2+10x+25+y^2+10y+25=y^2-14y+49$$

Combining like terms, we obtain:

$$x^2+10x+1+24y=0$$

Solving for $y$, we get the quadratic function:

$$y=-\frac{x^2+10x+1}{24}$$
 
Hello, Abs!

Find the equation of the parabola with focus at (-5, -5)
and directrix y = 7.
Code:
                    |
                    |7
          - - . - - + - - -
              :     |
              :V    |
              o     |
    - - - * - : - * + - - - - -
        *     :     *
       *      o     |*
              :F    |
      *       :     | *
                    |
The focus (F) is (-5,-5).
The vertex (V) is (-5,1).

The form of this parabola is: (x-h)^2 \:=\:4p(y-k)
where (h,k) is the vertex,
and p is the directed distance from V to F.

We have: (h,k) = (-5,1) and p = -6.

The equation is: .(x+5)^2 \:=\:-24(y-1)
 
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...

Similar threads

Replies
6
Views
2K
Replies
11
Views
3K
Replies
1
Views
7K
Replies
4
Views
3K
Replies
2
Views
2K
Back
Top