Respuesta :
8x²+3y²=24
8x²+3y²-24=0
Let x be 1 ,
Therefore
8+3y²-24=0
3y²-16=0
3y²=16
y²=16/3
y=4/1.7
y=40/17
Take different values of x &y and plot a graph!!!
An ellipse is defined as the path traced by a point as it moves such that the sum of distance from two fixed location is a constant
i) The center of the ellipse, is (0, 0)
ii) The vertex of the ellipse are (-√3, 0), and (√3, 0)
iii) Covertex coordinates are (0, 2·√2), and (0, -2·√2)
iv) Length of the minor axis is 2·√3
v) Length of the major axis is 4·√2
vi) Length of the latus rectum is [tex]\underline {\dfrac{3 \cdot \sqrt{2} }{2}}[/tex]
vii) The equation of the directrix, is x = ± 8·√5/5
viii) The equation of the latus rectum = √5
The reason the above values are correct are as follows:
The given equation is presented as follows;
8·x² + 3·y² = 24
Required:
To find the center, vertex, covertex, length of minor axis, length of latus rectum, equation of directrices and latus rectum
Solution;
i) The general form of the equation of an ellipse is presented as follows;
[tex]\dfrac{(x- h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1[/tex]
From the given equation, we get;
The center of the
[tex]\dfrac{8\cdot x^2}{24} + \dfrac{3\cdot y^2}{24} =\dfrac{x^2}{3} + \dfrac{y^2}{8} = 1[/tex]
The equation of the given ellipse, is [tex]\dfrac{x^2}{3} + \dfrac{y^2}{8} = 1[/tex]
∴ a = √3, b = 2·√2
The value of h, and k, are both 0
Therefore, the center of the ellipse, (h, k) = (0, 0)
ii) The vertex are (h - a, k), and (h + a, k), which gives;
The vertex are (0 - √3, 0), and (0 + √3, 0)
The vertex of the ellipse are (-√3, 0), and (√3, 0)
iii) The coordinates of the covertex of an ellipse are;
(h, k + b), and (h, k - b)
Therefore, the coordinates of the covertex of the given ellipse are;
(0, 0 + 2·√2), and (0, 0 - 2·√2), which gives;
The covertex coordinates = (0, 2·√2), and (0, -2·√2)
iv) The length of the minor axis = 2×a
∴ The length of the minor axis = 2·√3
v) The length of the major axis = 2 × b
∴ The length of the major axis, = 2 × 2·√2 = 4·√2
vi) The length of the latus rectum, LR, is given as follows;
b > a, therefore;
[tex]LR = \dfrac{2 \cdot a^2}{b}[/tex]
[tex]LR = \dfrac{2 \times 3}{2 \cdot \sqrt{2} } = \mathbf{ \dfrac{3 \cdot \sqrt{2} }{2}}[/tex]
vii) Equation of directrices
a² = b²·(1 - e²)
3 = 8·(1 - e²)
e² = 1 - 3/8 = 5/8
The equation of the directrix, x = ± a/e
The equation of the directrix, x = ±(2·√2)/√(5/8) = ± 8·√5/5
viii) The focus of the ellipse, C² = 8 - 3 = 5
The equation of the latus rectum = b·e = √8 × √(5/8) = √5
Learn more about an ellipse here:
ttps://brainly.com/question/22404367
