Find Equation of Ellipse | Eccentricity 2/3 | (2,0) Focus & x+y=0 Directrix

In summary: We can find the equation of the ellipse without knowing either of their values. The equation of the ellipse is given by:(x-2)^2 + y^2 = 4/9 ( ( (x+y)/√2)^2)
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Find the equation of the ellipse whose eccentricity is 2/3 and which has (2,0) and x+y=0 for focus and corresponding directrix .

given answer: (x-2)^2 + y^2 = 4/9 ( ( (x+y)/√2)^2)what i tried doing:-

ae=2
⇒a*2/3=2
∴a=3

found b=√5.

what to do next? please help.
 
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  • #2
Yatin said:
Find the equation of the ellipse whose eccentricity is 2/3 and which has (2,0) and x+y=0 for focus and corresponding directrix .

given answer: (x-2)^2 + y^2 = 4/9 ( ( (x+y)/√2)^2)what i tried doing:-

ae=2
⇒a*2/3=2
∴a=3

found b=√5.

what to do next? please help.
Have you drawn a sketch of the ellipse? Based on the given information, the major axis of the ellipse is a line with slope 1 that goes through the point (2, 0). You will need to know how to write the equation of an ellipse that is not in standard orientation. The rotation is the reason for the xy term in the equation.

Also, in future posts, please do not delete the three parts of the homework template.
 
  • #3
Mark44 said:
Have you drawn a sketch of the ellipse? Based on the given information, the major axis of the ellipse is a line with slope 1 that goes through the point (2, 0). You will need to know how to write the equation of an ellipse that is not in standard orientation. The rotation is the reason for the xy term in the equation.

Also, in future posts, please do not delete the three parts of the homework template.

Thanks a lot Mark.
 
  • #4
You started by calculating a value of some number, "a". Why? What does "a" mean? The fact that you did that must mean that you have some formula that involves "a". What is that formula?
 
  • #5
HallsofIvy said:
You started by calculating a value of some number, "a". Why? What does "a" mean? The fact that you did that must mean that you have some formula that involves "a". What is that formula?

Thanks for taking interest in my thread HallsofIvy.

'a' represents the semimajor axis of the ellipse.
'b' represents the semiminor axis.
e: eccentricity
Foci : + or - ae ( since there r 2 foci.)

I realized that finding a or b, however, was unnecessary.
 
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1. What is the equation of an ellipse with eccentricity 2/3, a focus at (2,0), and a directrix of x+y=0?

The equation of an ellipse with these characteristics is:
(x-2)^2 / (1 - (2/3)^2) + y^2 / (1 - (2/3)^2) = 1

2. How does the eccentricity affect the shape of an ellipse?

The eccentricity of an ellipse determines how elongated or circular the shape is. An eccentricity of 0 represents a perfect circle, while an eccentricity of 1 represents a line segment. In this case, an eccentricity of 2/3 indicates a moderately elongated ellipse.

3. What is the significance of the focus and directrix in an ellipse?

The focus and directrix are important elements of an ellipse, as they help define its shape. The focus is a fixed point inside the ellipse, while the directrix is a fixed line outside the ellipse. The distance from any point on the ellipse to the focus is always less than the distance to the directrix, and this relationship is what creates the characteristic shape of an ellipse.

4. Can an ellipse have more than one focus and directrix?

No, an ellipse can only have one focus and one directrix. These elements are unique to an ellipse and help to define its shape.

5. How is the equation of an ellipse with eccentricity 2/3 derived?

The equation of an ellipse with eccentricity 2/3 is derived using the distance formula and the definition of eccentricity. The distance between any point (x,y) on the ellipse and the focus (2,0) is equal to the distance between that point and the directrix x+y=0, multiplied by the eccentricity 2/3. This relationship can be algebraically manipulated to obtain the equation of the ellipse.

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