Advertisements
Advertisements
प्रश्न
Find the equation of the ellipse in standard form if passing through the points (−3, 1) and (2, −2)
Advertisements
उत्तर
Let the required equation of ellipse be `x^2/"a"^2 + y^2/"b"^2` = 1, where a > b.
The ellipse passes through the points (–3, 1) and (2, –2).
∴ Substituting x = –3 and y = 1 in equation of ellipse, we get
`(-3)^2/"a"^2 + 1^2/"b"^2` = 1
∴ `9/"a"^2 + 1/"b"^2` = 1 ...(i)
Substituting x = 2 and y = –2 in equation of ellipse, we get
`2^2/"a"^2 + (-2)^2/"b"^2` = 1
∴ `4/"a"^2 + 4/"b"^2` = 1 ...(ii)
Let `1/"a"^2` = A and `1/"b"^2` = B
∴ Equations (i) and (ii) become
9A + B = 1 …(iii)
4A + 4B = 1 …(iv)
Multiplying (iii) by 4, we get
36A + 4B = 4 …(v)
Subtracting (iv) from (v), we get
32A = 3
∴ A = `3/32`
Substituting A = `3/32` in (iv), we get
`4(3/32) + 4"B"` = 1
∴ `3/8 + 4"B"` = 1
∴ 4B = `1 - 3/8`
∴ 4B = `5/8`
∴ B = `5/32`
Since `1/"a"^2` = A and `1/"b"^2` = B,
`1/"a"^2 = 3/32` and `1/"b"^2 = 5/32`
∴ a2 = `32/3` and b2 = `32/5`
∴ The required equation of ellipse is
`x^2/((32/3)) + y^2/((32/5))`, i.e., 3x2 + 5y2 = 32.
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
- 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 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 eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.
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
Show that the line x – y = 5 is a tangent to the ellipse 9x2 + 16y2 = 144. Find the point 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 5x2 + 9y2 = 45 which are ⊥ to the line 3x + 2y + y = 0.
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`.
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:
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 tangent to the ellipse 4x2 + 9y2 = 36 which is perpendicular to the 3x + 4y = 17 is,
Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8
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 ______.
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 ______.
The equation of the ellipse with its centre at (1, 2), one focus at (6, 2) and passing through the point (4, 6) is ______.
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 ______.
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 ______.
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______.
Length of latusrectum of the ellipse \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\] is______.
The distance between the foci of the ellipse \[3x^2+4y^2=48\] is______.
