Advertisements
Advertisements
Question
Find the equation of the ellipse in standard form if eccentricity is `2/3` and passes through `(2, −5/3)`.
Advertisements
Solution
Let the required equation of ellipse be `x^2/"a"^2 + y^2/"b"^2` = 1, where a > b.
Given, eccentricity (e) = `2/3`
The ellipse passes through `(2, -5/3)`.
∴ Substituting x = 2 and y = `-5/3` in equation of ellipse, we get
`2^2/"a"^2 + (-5/3)^2/"b"^2` = 1
∴ `4/"a"^2 + 25/(9"b"^2)` = 1
∴ `4/"a"^2 + 25/(9"a"^2(1 - "e"^2)` = 1 ...[∵ b2 = a2 (1 – e2)]
Multiplying throughout by a2, we get
`4 + 25/(9(1 - "e"^2)` = a2
∴ `4 + 25/(9[1 - (2/3)^2])` = a2
∴ `4 + 25/(9(1 - 4/9)` = a2
∴ `4 + 25/5` = a2
∴ 4 + 5 = a2
∴ a2 = 9
Now, b2 = a2 (1 – e2)
= `9[1 - (2/3)^2]`
= `9(1 - 4/9)`
= `9(5/9)`
= 5
∴ The required equation of ellipse is `x^2/9 + y^2/5` = 1.
APPEARS IN
RELATED QUESTIONS
Find the
- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrics
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:
3x2 + 4y2 = 12
Find the
- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrics
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:
2x2 + 6y2 = 6
Find the
- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrices
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:
3x2 + 4y2 = 1
Find the equation of the ellipse in standard form if eccentricity = `3/8` and distance between its foci = 6
Find the equation of the ellipse in standard form if the distance between directrix is 18 and eccentricity is `1/3`.
Find the equation of the ellipse in standard form if the minor axis is 16 and eccentricity is `1/3`.
Find the equation of the ellipse in standard form if the latus rectum has length of 6 and foci are (±2, 0).
Find the equation of the ellipse in standard form if passing through the points (−3, 1) and (2, −2)
Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.
A tangent having slope `–1/2` to the ellipse 3x2 + 4y2 = 12 intersects the X and Y axes in the points A and B respectively. If O is the origin, find the area of the triangle
Determine whether the line `x + 3ysqrt(2)` = 9 is a tangent to the ellipse `x^2/9 + y^2/4` = 1. If so, find the co-ordinates of the pt of contact
Find the equation of the tangent to the ellipse 4x2 + 7y2 = 28 from the point (3, –2).
Find the equation of the tangent to the ellipse `x^2/25 + y^2/4` = 1 which are parallel to the line x + y + 1 = 0.
Find the equation of the locus of a point the tangents form which to the ellipse 3x2 + 5y2 = 15 are at right angles
Tangents are drawn through a point P to the ellipse 4x2 + 5y2 = 20 having inclinations θ1 and θ2 such that tan θ1 + tan θ2 = 2. Find the equation of the locus of P.
The eccentric angles of two points P and Q the ellipse 4x2 + y2 = 4 differ by `(2pi)/3`. Show that the locus of the point of intersection of the tangents at P and Q is the ellipse 4x2 + y2 = 16
Select the correct option from the given alternatives:
The equation of the ellipse is 16x2 + 25y2 = 400. The equations of the tangents making an angle of 180° with the major axis are
Select the correct option from the given alternatives:
The equation of the tangent to the ellipse 4x2 + 9y2 = 36 which is perpendicular to the 3x + 4y = 17 is,
Select the correct option from the given alternatives:
Centre of the ellipse 9x2 + 5y2 − 36x − 50y − 164 = 0 is at
Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8
On the ellipse `x^2/8 + "y"^2/4` = 1 let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S' be the foci of the ellipse and e be its eccentricity. If A is the area of the triangle SPS' then, the value of (5 – e2). A is ______.
If the tangents on the ellipse 4x2 + y2 = 8 at the points (1, 2) and (a, b) are perpendicular to each other, then a2 is equal to ______.
The equation of the ellipse with its centre at (1, 2), one focus at (6, 2) and passing through the point (4, 6) is ______.
The point on the ellipse x2 + 2y2 = 6 closest to the line x + y = 7 is (a, b). The value of (a + b) will be ______.
The ratio of the area of the ellipse and the area enclosed by the locus of mid-point of PS where P is any point on the ellipse and S is the focus of the ellipse, is equal to ______.
Eccentricity of ellipse `x^2/a^2 + y^2/b^2` = 1, if it passes through point (9, 5) and (12, 4) is ______.
Equation of the ellipse whose axes are along the coordinate axes, vertices are (± 5, 0) and foci at (± 4, 0) is ______.
A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is `1/2`. Then the length of the semi-major axis is ______.
