is the equation of an ellipse with directrices at x = ±4 and foci at (2, 0) and (−2, 0).
It was given that, the ellipse has directrices at
The directrices has equation:
Where, e is the eccentricity of the ellipse.
Given the foci are (2, 0) and (−2, 0). It denotes (a e, 0) and (- a e, 0). Hence,
a e = 2 >eq.2
Multiply eq. 1 and 2, we get
Taking square root, we get
Now substitute this ‘a’ value in eq. 2, we get
Substituting the values, we get
The equation for an ellipse is
x squared over 8 plus y squared over 4 equals 1
The general equation of ellipse is given by :-
For foci =(c,0) , and a>b we have .
Given, For an ellipse , we have
Directrix at x = ±4 and foci at (2, 0) and (−2, 0).
Since, directrix = and , where e is the eccentricity.
So, the equation of ellipse becomes
hence, the correct option is .
Know the vocabulary for algebraic fractions. The following terms will be used throughout the examples, and are common in problems involving algebraic fractions:
Numerator: The top part of a fraction (ie. (x+5)/(2x+3)).
Denominator: The bottom part of the fraction (ie. (x+5)/(2x+3)).
Common Denominator: This is a number that you can divide out of both the top and bottom of a fraction. For example, in the fraction 3/9, the common denominator is 3, since both numbers can be divided by 3.
Factor: One number that multiples to make another. For example, the factors of 15 are 1, 3, 5, and 15. The factors of 4 are 1, 2, and 4.
Simplified Equation: This involves removing all common factors and grouping similar variables together (5x + x = 6x) until you have the most basic form of a fraction, equation, or problem. If you cannot do anything more to the fraction, it is simplified.
Review how to solve simple fractions. These are the exact same steps you will take to solve algebraic fractions. Take the example, 15/35. In order to simplify a fraction, we need to find a common denominator. In this case, both numbers can be divided by five, so you can remove the 5 from the fraction:
15 → 5 * 3
35 → 5 * 7
Now you can cross out like terms. In this case you can cross out the two fives, leaving your simplified answer, 3/7.
Remove factors from algebraic expressions just like normal numbers. In the previous example, you could easily remove the 5 from 15, and the same principle applies to more complex expressions like, 15x – 5. Find a factor that both numbers have in common. Here, the answer is 5, since you can divide both 15x and -5 by the number five. Like before, remove the common factor and multiply it by what is “left.”
15x – 5 = 5 * (3x – 1)
To check your work, simply multiply the five back into the new expression – you will end up with the same numbers you started with.
I hope this helped : )
x^2/8 + y^2/4 = 1
As the diretrix are vertical lines, we have a horizontal ellipse, which equation is:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
As the foci are at (2,0) and (-2,0), we have that k = 0, h = 2-2 = 0 and c = 2, where c^2 = a^2 - b^2
As the diretrix are in x = ±4, we have that d = 4, where:
c / a = a / d
So now we can find a:
2 / a = a / 4
a^2 = 8
a = 2.828
And then we can find b:
2^2 = 2.828^2 - b^2
b^2 = 2.828^2 - 2^2 = 4
b = 2
So the ellipse equation is:
x^2/8 + y^2/4 = 1