Advertisements
Advertisements
प्रश्न
Identify the type of conic and find centre, foci, vertices, and directrices of the following:
18x2 + 12y2 – 144x + 48y + 120 = 0
उत्तर
18x2 – 144x + 12y2 + 48y = – 120
18(x2 – 8x) + 12(y2 + 4y) = – 120
18(x – 4)2 + 12(y + 2)2 = – 120 + 288 + 48
18(x – 4)2 + 12(y + 2)2 = 216
`(18(x - 4)^2)/216 + (12(y + 2)^2)/216` = 1
`(x - 4)^2/12 + (y + 2)^2/18` = 1
It is an ellipse.
The major axis is parallell to y axis.
a2 = 18, b2 = 12
a = `sqrt(18), "b" = 2sqrt(3)`
= `3sqrt(2)`
c2 = a2 + b2
= 18 – 12
= 6
c = `sqrt(6)`
ae = `sqrt(6)`
`3sqrt(2)"e" = sqrt(6)`
e = `sqrt(6)/(3sqrt(2)) = sqrt(3)/3 = 1/sqrt(3)`
Centre (h, k) = (4, – 2)
Vertices (h, ±a + k) = `(4, +- 3sqrt(2) - 2)`
= `(4, 3sqrt(2) - 2)` and `(4, -3sqrt(2) - 2)`
Foci (h, ±c + k) = `(4, +- sqrt(6) - 2)`
= `(4 +- sqrt(6) - 2)` and `(4, - sqrt(6) - 2)`
Directrix y = `+- "a"/"e" + "k"`
= `+- (3sqrt(2))/1 - 2`
= `+- 3sqrt(6) - 2`
y = `3sqrt(6) - 2` and y = `- 3sqrt(6) - 2`
APPEARS IN
संबंधित प्रश्न
Find the co-ordinates of the focus, vertex, equation of the directrix, axis and the length of latus rectum of the parabola
x2 = 8y
The average variable cost of the monthly output of x tonnes of a firm producing a valuable metal is ₹ `1/5`x2 – 6x + 100. Show that the average variable cost curve is a parabola. Also, find the output and the average cost at the vertex of the parabola.
The profit ₹ y accumulated in thousand in x months is given by y = -x2 + 10x – 15. Find the best time to end the project.
Find the equation of the parabola which is symmetrical about x-axis and passing through (–2, –3).
Find the axis, vertex, focus, equation of directrix and the length of latus rectum of the parabola (y - 2)2 = 4(x - 1)
The focus of the parabola x2 = 16y is:
The eccentricity of the parabola is:
Find the equation of the ellipse in the cases given below:
Foci `(+- 3, 0), "e"+ 1/2`
Find the equation of the ellipse in the cases given below:
Foci (0, ±4) and end points of major axis are (0, ±5)
Find the equation of the ellipse in the cases given below:
Length of latus rectum 4, distance between foci `4sqrt(2)`, centre (0, 0) and major axis as y-axis
Find the equation of the hyperbola in the cases given below:
Foci (± 2, 0), Eccentricity = `3/2`
Find the equation of the hyperbola in the cases given below:
Centre (2, 1), one of the foci (8, 1) and corresponding directrix x = 4
Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
y2 = – 8x
Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
y2 – 4y – 8x + 12 = 0
Identify the type of conic and find centre, foci, vertices, and directrices of the following:
`x^2/25 + y^2/9` = 1
Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis
Identify the type of conic and find centre, foci, vertices, and directrices of the following:
`(x + 1)^2/100 + (y - 2)^2/64` = 1
