Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
The equation of the ellipse is 3x2 + 4y2 = 1
i.e., `x^2/((1/3)) + y^2/((1/4))` = 1
Comparing with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
a2 = `1/3`, b2 = `1/4`
∴ a = `1/sqrt3`, b = `1/2`
∴ a > b
i. Length of major axis = 2a = `2/sqrt(3)`
Length of minor axis = 2b = `2(1/2)` = 1
ii. Eccentricity = e = `sqrt("a"^2 - "b"^2)/"a"`
= `sqrt(1/3 - 1/4)/((1/sqrt(3))`
= `((1/(2sqrt(3))))/((1/sqrt(3))) = 1/2`
∴ ae = `1/sqrt(3) xx 1/2 = 1/(2sqrt(3))`
∴ coordinates of foci = (± ae, 0) = `(± 1/(2sqrt(3)), 0)`.
iii. `"a"/"e" = ((1/sqrt(3)))/((1/2)) = 2/sqrt(3)`
The equations of directrices are
x = `± "a"/"e"`
∴ x = `± 2/sqrt(3)`
iv. Length of latus rectum = `(2"b"^2)/"a"`
= `(2 xx 1/4)/((1/sqrt(3))`
= `sqrt(3)/2`
v. Distance between foci = 2ae
= `2 xx 1/(2sqrt(3)`
= `1/sqrt(3)`
vi. Distance between directrices = `(2"a")/"e"`
= `2 xx 2/sqrt(3)`
= `4/sqrt(3)`
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:
2x2 + 6y2 = 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 equation of the ellipse in standard form if eccentricity is `2/3` and passes through `(2, −5/3)`.
Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its foci
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 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 2x2 + y2 = 6 from the point (2, 1).
Find the equation of the tangent to the ellipse x2 + 4y2 = 20, ⊥ to the line 4x + 3y = 7.
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.
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 tangent to the ellipse 4x2 + 9y2 = 36 which is perpendicular to the 3x + 4y = 17 is,
The length of the latusrectum of an ellipse is `18/5` and eccentncity is `4/5`, then equation of the ellipse is ______.
Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of `π/2` at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse E: `x^2/a^2 + y^2/b^2` = 1, a2 > b2. If e is the eccentricity of the ellipse E, then the value of `1/e^2` is equal to ______.
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 ______.
An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms, are ______.
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.
Let the ellipse `x^2/a^2 + y^2/b^2` = 1 has latus sectum equal 8 units – if the ellipse passes through `(sqrt(5), 4)` Then The radius of the directive circle is ______.
The points where the normals to the ellipse x2 + 3y2 = 37 are parallel to the line 6x – 5y = 2 are ______.
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 ______.
Let the eccentricity of an ellipse `x^2/a^2 + y^2/b^2` = 1, a > b, be `1/4`. If this ellipse passes through the point ```(-4sqrt(2/5), 3)`, then a2 + b2 is equal to ______.
If P1 and P2 are two points on the ellipse `x^2/4 + y^2` = 1 at which the tangents are parallel to the chord joining the points (0, 1) and (2, 0), then the distance between P1 and P2 is ______.
Eccentricity of ellipse `x^2/a^2 + y^2/b^2` = 1, if it passes through point (9, 5) and (12, 4) is ______.
The locus of a variable point whose distance from (- 2, 0) is \[\frac{2}{3}\] times its distance from the line \[x=-\frac{9}{2}\], is______.
If the length of the major axis of the ellipse \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] is three times the length of minor axis, then its eccentricity is______.
The distance between the foci of the ellipse \[x=3\text{cos}\theta,y=4\text{sin}\theta\] is______.
