Advertisements
Advertisements
प्रश्न
Find the equation of the ellipse in standard form if the latus rectum has length of 6 and foci are (±2, 0).
Advertisements
उत्तर
Let the equation of the ellipse be
`x^2/"a"^2 + y^2/"b"^2` = 1 ...(1)
Then length of latus rectum = `(2"b"^2)/"a"`
and foci = (±ae, 0)
∴ `(2"b"^2)/"a"` = 6 ...(2)
ae = 2 ...(3)
∵ b2 = a2 (1 – e2), from (2),
`(2"a"^2(1 - "e"^2))/"a"` = 6
∴ (a2 – a2e2) = 3a
∴ a2 – 4 = 3a ....[By (3)]
∴ a2 – 3a – 4 = 0
∴ (a – 4)(a + 1) = 0
∴ a = 4 or a = – 1
But a ≠ – 1
∴ a = 4
∴ from (2), `(2"b"^2)/4` = 6
∴ b2 = 12
∴ from (1), the equation of the required ellipse is
`x^2/16 + y^2/12` = 1
APPEARS IN
संबंधित प्रश्न
Answer the following:
Find the
- lengths of the principal axes
- co-ordinates of the foci
- equations of directrices
- length of the latus rectum
- distance between foci
- distance between directrices of the ellipse:
`x^2/25 + y^2/9` = 1
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 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)`.
Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its foci
Find k, if the line 3x + 4y + k = 0 touches 9x2 + 16y2 = 144
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
Find the equations of the tangents to the ellipse `x^2/16 + y^2/9` = 1, making equal intercepts on co-ordinate axes
Select the correct option from the given alternatives:
The equation of the ellipse having eccentricity `sqrt(3)/2` and passing through (− 8, 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,
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 ______.
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 ______.
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 ______.
The tangent and the normal at a point P on an ellipse `x^2/a^2 + y^2/b^2` = 1 meet its major axis in T and T' so that TT' = a then e2cos2θ + cosθ (where e is the eccentricity of the ellipse) 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 ______.
The eccentricity, foci and the length of the latus rectum of the ellipse x2 + 4y2 + 8y – 2x + 1 = 0 are respectively equal to ______.
If the chord through the points whose eccentric angles are α and β on the ellipse `x^2/a^2 + y^2/b^2` = 1 passes through the focus (ae, 0), then the value of tan `α/2 tan β/2` will be ______.
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 ______.
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 ______.
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 ______.
\[\frac{x^{2}}{r^{2}-r-6}+\frac{y^{2}}{r^{2}-6r+5}=1\] will represent the ellipse, if r lies in the interval______.
Eccentricity of the conic \[16x^2+7y^2=112\] is______.
For the ellipse \[3x^2+4y^2=12,\] the length of latus rectum is______.
