1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
emmainna [20.7K]
3 years ago
11

A parabola has its focus at (1,2) and its directrix is y=-2. the equation of this parabola could be

Mathematics
1 answer:
Oksi-84 [34.3K]3 years ago
4 0
<span>x^2/8 - x/4 + 1/8 = 0 A parabola is defined as the set of all points such that each point has the same distance from the focus and the directrix. Also the parabola's equation will be a quadratic equation of the form ax^2 + bx + c. So if we can determine 3 points on the parabola, we can use those points to calculate the desired equation. First, let's draw the shortest possible line from the focus to the directrix. The midpoint of that line will be a point on the desired parabola. Since the slope of the directrix is 0, the line will have the equation of x=1. This line segment will be from (1,2) to (1,-2) and the midpoint will be ((1+1)/2, (2 + -2)/2) = (2/2, 0/2) = (1,0). Now for the 2nd point, let's draw a line that's parallel to the directrix and passing through the focus. The equation of that line will be y=2. Any point on that line will have a distance of 4 from the directrix. So let's give it an x-coordinate value of (1+4) = 5. So another point for the parabola is (5,2). And finally, if we subtract 4 instead of adding 4 to the x coordinate, we can get a third point of 1-4 = -3. So that 3rd point is (-3,2). So we now have 3 points on the parabola. They are (1,0), (5,2), and (-3,2). Let's create some equations of the form ax^2 + bx + c = y and then substitute the known values into those equations. SO ax^2 + bx + c = y (1) a*1^2 + b*1 + c = 0 (2) a*5^2 + b*5 + c = 2 (3) a*(-3)^2 + b*(-3) + c = 2 Let's do the multiplication for those expressions. So (4) a + b + c = 0 (5) 25a + 5b + c = 2 (6) 9a - 3b + c = 2 Equations (5) and (6) above look interesting. Let's subtract (6) from (5). So 25a + 5b + c = 2 - 9a - 3b + c = 2 = 16a + 8b = 0 Now let's express a in terms of b. 16a + 8b = 0 16a = -8b a = -8b/16 (7) a = -b/2 Now let's substitute the value (-b/2) for a in expression (4) above. So a + b + c = 0 -b/2 + b + c = 0 And solve for c -b/2 + b + c = 0 b/2 + c = 0 (8) c = -b/2 So we know that a = -b/2 and c = -b/2. Let's substitute those values for a and c in equation (5) above and solve for b. 25a + 5b + c = 2 25(-b/2) + 5b - b/2 = 2 -25b/2 + 5b - b/2 = 2 2(-25b/2 + 5b - b/2) = 2*2 -25b + 10b - b = 4 -16b = 4 b = -4/16 b = -1/4 So we now know that b = -1/4. Using equations (7) and (8) above, let's calculate a and c. a = -b/2 = -(-1/4)/2 = 1/4 * 1/2 = 1/8 c = -b/2 = -(-1/4)/2 = 1/4 * 1/2 = 1/8 So both a and c are 1/8. So the equation for the parabola is x^2/8 - x/4 + 1/8 = 0 Let's test to make sure it works. First, let's use an x of 1. x^2/8 - x/4 + 1/8 = y 1^2/8 - 1/4 + 1/8 = y 1/8 - 1/4 + 1/8 = y 1/8 - 2/8 + 1/8 = y 0 = y And we get 0 as expected. Let's try x = 2 x^2/8 - x/4 + 1/8 = y 2^2/8 - 2/4 + 1/8 = y 4/8 - 1/2 + 1/8 = y 4/8 - 1/2 + 1/8 = y 1/2 - 1/2 + 1/8 = y 1/8 = y. Let's test if (2,1/8) is the same distance from both the focus and the directrix. The distance from the directrix is 1/8 - (-2) = 1/8 + 2 = 1/8 + 16/8 = 17/8 The distance from the focus is d = sqrt((2-1)^2 + (1/8-2)^2) d = sqrt(1^2 + -15/8^2) d = sqrt(1 + 225/64) d = sqrt(289/64) d = 17/8 And the distances match again. So we do have the correct equation of: x^2/8 - x/4 + 1/8 = 0</span>
You might be interested in
Slope of the line passing through points (-6,6) and (4,-7)
castortr0y [4]
The slope of a line is constant because the acceleration is equal to zero.

The slope of a line is always just (y2-y1)/(x2-x1), so in this case:

m=(-7-6)/(4--6)

m=-13/10

m=-1.3
3 0
3 years ago
A cost regularly priced at $188 is on sale for 25% off . What is the sale price ?
Brums [2.3K]
You need to multiply 188 by 25%. But first convert 25% to a decimal 0.25.
188*0.25 is 47
Now we subtract 188 - 47
Which is 141
$141 is the sales price.
5 0
3 years ago
Read 2 more answers
Can someone help me with this?- Its the end of the year word search for math and its so hard :(
Darina [25.2K]
In the 8th row there’s a Q and if u go sideways down it’s quadrilateral if u need more lmk!
7 0
3 years ago
Find the mean of the following data: 10, 16, 15, 14, 8, 21, 10, 5, 19, 18, 4, 5, 16, 12, 10, 9
Evgesh-ka [11]

Answer:

12

Step-by-step explanation:

add all them together, then divide by the amt. of numbers there are.

5 0
3 years ago
Read 2 more answers
Solve<br> 2(x+3)=x-4<br><br> im not sure what to do with this question
vovangra [49]

Answer:

x = -10

Step-by-step explanation:

2(x + 3) = x - 4

2x + 6 = x - 4

2x - x = -4 - 6

x = -10

3 0
3 years ago
Other questions:
  • In circle W, if VU XS, find XS
    14·1 answer
  • Alexa is designing a paper airplane whose final shape, when viewed from the top or bottom, is a trapezoid. A sketch of her
    13·1 answer
  • Cady feeds her cats Cat Crunchies. Each day Sammy eats 1/2 of the box, tommy eats 1/8 of the box, and Suzi eats 1/4 of the box .
    13·2 answers
  • What are three collinear points on line l?
    7·1 answer
  • Find the measure of angle x in the figure below:<br><br> 35°<br> 47°<br> 73°<br> 78°
    14·2 answers
  • Which of the following is equivalent to 8 × 8/6?
    12·1 answer
  • There are 10 students in a class. The teacher chooses 3 students to go to the
    11·1 answer
  • For people who have taken PreCalc/Calc:
    9·2 answers
  • ILL GIVE BRAINLIEST!
    5·1 answer
  • What is the probability that when you roll the number cube (6 sides) twice, you will get 1 on both rolls?
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!