Advertisements
Advertisements
Question
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
36x2 + 4y2 = 144
Advertisements
Solution
Equation of ellipse 36x2 + 4y2 = 144
`(36x^2)/144 + (4y^2)/144 = 1` `x^2/4 + y^2/36 = 1`
∴ a2 = 36, b2 = 4
∴ a = 6, b = 2
∴ c2 = a2 – b2 = 36 – 4 = 32
∴ c = `4sqrt2`
The axis of the ellipse is along the y-axis
Coordinates of foci are (0, ± c) or (0, ± `4sqrt2`
Coordinates of vertices are (0, ± a) or (0, ± 6)
Length of major axis = 2a = 2 × 6 = 12
Length of minor axis = 2b = 2 × 2 = 4
Eccentricity (e) = `c/a`
= `(4sqrt2)/6`
= `(2sqrt2)/3`
Length of latus rectum = `(2"b")^2/a`
= `(2 xx 4)/6`
= `4/3`
APPEARS IN
RELATED QUESTIONS
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/25 + y^2/100 = 1`
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/49 + y^2/36 = 1`
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/100 + y^2/400 = 1`
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
16x2 + y2 = 16
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
4x2 + 9y2 = 36
An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola:
y2 = 8x
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
4x2 + y = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas
y2 − 4y − 3x + 1 = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 − 4y + 4x = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 + 4x + 4y − 3 = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 = 8x + 8y
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 = 8x + 8y
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
4 (y − 1)2 = − 7 (x − 3)
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 = 5x − 4y − 9
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
x2 + y = 6x − 14
For the parabola y2 = 4px find the extremities of a double ordinate of length 8 p. Prove that the lines from the vertex to its extremities are at right angles.
Find the length of the line segment joining the vertex of the parabola y2 = 4ax and a point on the parabola where the line-segment makes an angle θ to the x-axis.
Write the length of the chord of the parabola y2 = 4ax which passes through the vertex and is inclined to the axis at \[\frac{\pi}{4}\]
Write the coordinates of the vertex of the parabola whose focus is at (−2, 1) and directrix is the line x + y − 3 = 0.
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
x2 + 2y2 − 2x + 12y + 10 = 0
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
x2 + 4y2 − 4x + 24y + 31 = 0
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
4x2 + 16y2 − 24x − 32y − 12 = 0
Find the equation of an ellipse whose foci are at (± 3, 0) and which passes through (4, 1).
A rod of length 12 m moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with x-axis.
If the lengths of semi-major and semi-minor axes of an ellipse are 2 and \[\sqrt{3}\] and their corresponding equations are y − 5 = 0 and x + 3 = 0, then write the equation of the ellipse.
PSQ is a focal chord of the ellipse 4x2 + 9y2 = 36 such that SP = 4. If S' is the another focus, write the value of S'Q.
If a latus rectum of an ellipse subtends a right angle at the centre of the ellipse, then write the eccentricity of the ellipse.
Find the equation of the ellipse with foci at (± 5, 0) and x = `36/5` as one of the directrices.
The equation of the circle having centre (1, –2) and passing through the point of intersection of the lines 3x + y = 14 and 2x + 5y = 18 is ______.
The equation of the ellipse whose centre is at the origin and the x-axis, the major axis, which passes through the points (–3, 1) and (2, –2) is ______.
Find the equation of a circle which touches both the axes and the line 3x – 4y + 8 = 0 and lies in the third quadrant.
Find the distance between the directrices of the ellipse `x^2/36 + y^2/20` = 1
