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.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.
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.
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?
The equation of an ellipse with these characteristics is:
(x-2)^2 / (1 - (2/3)^2) + y^2 / (1 - (2/3)^2) = 1
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.
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.
No, an ellipse can only have one focus and one directrix. These elements are unique to an ellipse and help to define its shape.
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.