Advertisements
Advertisements
प्रश्न
Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.
उत्तर
Let the equation of the ellipse be
`x^2/"a"^2 + y^2/"b"^2` = 1
It is given that,
l(LR) = `1/3l("minor axis")`
∴ `(2"b"^2)/"a" = 1/3(2"b")`
∴ 3b = a
∴ 9b2 = a2
∴ 9a2(1 – e2) = a2
∴ 9(1 – e2) = 1
∴ 9 – 9e2 = 1
∴ 8 = 9e2
∴ e2 = `8/9`
∴ e = `(2sqrt(2))/3` ... [∵ 0 < e < 1]
APPEARS IN
संबंधित प्रश्न
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 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 foci is 6 and the distance between directrix is `50/3`.
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 equation of the ellipse in standard form if the dist. between its directrix is 10 and which passes through `(-sqrt(5), 2)`.
Find the equation of the ellipse in standard form if eccentricity is `2/3` and passes through `(2, −5/3)`.
Show that the product of the lengths of the perpendicular segments drawn from the foci to any tangent line to the ellipse `x^2/25 + y^2/16` = 1 is equal to 16
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 k, if the line 3x + 4y + k = 0 touches 9x2 + 16y2 = 144
Find the equation of the tangent to the ellipse `x^2/5 + y^2/4` = 1 passing through the point (2, –2)
Find the equation of the tangent to the ellipse 4x2 + 7y2 = 28 from the point (3, –2).
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.
Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse
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:
If `"P"(pi/4)` is any point on he ellipse 9x2 + 25y2 = 225. S and S1 are its foci then SP.S1P =
Select the correct option from the given alternatives:
The equation of the ellipse having foci (+4, 0) and eccentricity `1/3` is
Select the correct option from the given alternatives:
If the line 4x − 3y + k = 0 touches the ellipse 5x2 + 9y2 = 45 then the value of k is
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
Answer the following:
Find the equation of the tangent to the ellipse x2 + 4y2 = 100 at (8, 3)
The length of the latusrectum of an ellipse is `18/5` and eccentncity is `4/5`, then equation of the ellipse is ______.
The eccentricity, foci and the length of the latus rectum of the ellipse x2 + 4y2 + 8y – 2x + 1 = 0 are respectively equal to ______.
Tangents are drawn from a point on the circle x2 + y2 = 25 to the ellipse 9x2 + 16y2 – 144 = 0 then find the angle between the tangents.
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 points where the normals to the ellipse x2 + 3y2 = 37 are parallel to the line 6x – 5y = 2 are ______.
The normal to the ellipse `x^2/a^2 + y^2/b^2` = 1 at a point P(x1, y1) on it, meets the x-axis in G. PN is perpendicular to OX, where O is origin. Then value of ℓ(OG)/ℓ(ON) is ______.
Eccentricity of ellipse `x^2/a^2 + y^2/b^2` = 1, if it passes through point (9, 5) and (12, 4) 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 ______.
