Advertisements
Advertisements
Question
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
Advertisements
Solution
Given equation of the ellipse is 2x2 + 6y2 = 6.
∴ `x^2/3 + y^2/1` = 1
Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
a2 = 3 and b2 = 1
∴ a = `sqrt(3) and "b"` = 1
Since a > b,
X-axis is the major axis and Y-axis is the minor axis.
i. Length of major axis = 2a = `2sqrt(3)`
Length of minor axis = 2b = 2(1) = 2
∴ Lengths of the principal axes are `2sqrt(3)` and 2.
ii. We know that e = `sqrt("a"^2 - "b"^2)/"a"`
∴ e = `sqrt(3 - 1)/sqrt(3) = sqrt(2)/sqrt(3)`
Co-ordinates of the foci are S(ae, 0) and S'(–ae, 0),
i.e., `"S"(sqrt(3)(sqrt(2)/sqrt(3)),0)` and `"S""'"(-sqrt(3)(sqrt(2)/sqrt(3)),0)`,
i.e., `"S"(sqrt(2), 0)` and `"S""'"(-sqrt(2), 0)`
iii. Equations of the directrices are x = `±"a"/"e"`,
i.e., x = `±sqrt3/(sqrt2/sqrt3)`, i.e., x = `±3/sqrt(2)`
iv. Length of latus rectum = `(2"b"^2)/"a" = (2(1)^2)/sqrt(3) = 2/sqrt(3)`
v. Distance between foci = 2ae
= `2(sqrt(3)) (sqrt(2)/sqrt(3))`
= `2sqrt(2)`
vi. Distance between directrices = `(2"a")/"e"`
= `(2sqrt(3))/(sqrt(2)/sqrt(3))`
= `(2 xx 3)/sqrt(2)`
= `3sqrt(2)`
APPEARS IN
RELATED QUESTIONS
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 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 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
Show that the line x – y = 5 is a tangent to the ellipse 9x2 + 16y2 = 144. Find the point of contact
Show that the line 8y + x = 17 touches the ellipse x2 + 4y2 = 17. Find the point of contact
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 2x2 + y2 = 6 from the point (2, 1).
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.
P and Q are two points on the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 with eccentric angles θ1 and θ2. Find the equation of the locus of the point of intersection of the tangents at P and Q if θ1 + θ2 = `π/2`.
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:
Centre of the ellipse 9x2 + 5y2 − 36x − 50y − 164 = 0 is at
The length of the latusrectum of an ellipse is `18/5` and eccentncity is `4/5`, then equation of the ellipse is ______.
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 ______.
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 ______.
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 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 ______.
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 ______.
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 ______.
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______.
Eccentricity of the conic \[16x^2+7y^2=112\] is______.
The distance between the foci of the ellipse \[3x^2+4y^2=48\] is______.
