Page No 26.22:
Question 1:
Find the equation of the ellipse whose focus is (1, −2), the directrix 3x − 2y + 5 = 0 and eccentricity equal to 1/2.
Answer:
Page No 26.22:
Question 2:
Find the equation of the ellipse in the following cases:
(i) focus is (0, 1), directrix is x + y = 0 and e =
(ii) focus is (−1, 1), directrix is x − y + 3 = 0 and e =
(iii) focus is (−2, 3), directrix is 2x + 3y + 4 = 0 and e =
(iv) focus is (1, 2), directrix is 3x + 4y − 5 = 0 and e = .
Answer:
(i)
(ii)
(iii)
(iv)
Page No 26.22:
Question 3:
Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
(i) 4x2 + 9y2 = 1
(ii) 5x2 + 4y2 = 1
(iii) 4x2 + 3y2 = 1
(iv) 25x2 + 16y2 = 1600.
(v) 9x2 + 25y2 = 225
Answer:
Page No 26.22:
Question 4:
Find the equation to the ellipse (referred to its axes as the axes of x and y respectively) which passes through the point (−3, 1) and has eccentricity .
Answer:
Page No 26.22:
Question 5:
Find the equation of the ellipse in the following cases:
(i) eccentricity e = and foci (± 2, 0)
(ii) eccentricity e = and length of latus rectum = 5
(iii) eccentricity e = and semi-major axis = 4
(iv) eccentricity e = and major axis = 12
(v) The ellipse passes through (1, 4) and (−6, 1).
(vi) Vertices (± 5, 0), foci (± 4, 0)
(vii) Vertices (0, ± 13), foci (0, ± 5)
(viii) Vertices (± 6, 0), foci (± 4, 0)
(ix) Ends of major axis (± 3, 0), ends of minor axis (0, ± 2)
(x) Ends of major axis (0, ± ), ends of minor axis (± 1, 0)
(xi) Length of major axis 26, foci (± 5, 0)
(xii) Length of minor axis 16 foci (0, ± 6)
(xiii) Foci (± 3, 0), a = 4
Answer:
Page No 26.23:
Question 6:
Find the equation of the ellipse whose foci are (4, 0) and (−4, 0), eccentricity = 1/3.
Answer:
Page No 26.23:
Question 7:
Find the equation of the ellipse in the standard form whose minor axis is equal to the distance between foci and whose latus-rectum is 10.
Answer:
Page No 26.23:
Question 8:
Find the equation of the ellipse whose centre is (−2, 3) and whose semi-axis are 3 and 2 when major axis is (i) parallel to x-axis (ii) parallel to y-axis.
Answer:
Page No 26.23:
Question 9:
Find the eccentricity of an ellipse whose latus rectum is
(i) half of its minor axis
(ii) half of its major axis.
Answer:
Page No 26.23:
Question 10:
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
(i) x2 + 2y2 − 2x + 12y + 10 = 0
(ii) x2 + 4y2 − 4x + 24y + 31 = 0
(iii) 4x2 + y2 − 8x + 2y + 1 = 0
(iv) 3x2 + 4y2 − 12x − 8y + 4 = 0
(v) 4x2 + 16y2 − 24x − 32y − 12 = 0
(vi) x2 + 4y2 − 2x = 0
Answer:
Page No 26.23:
Question 11:
Find the equation of an ellipse whose foci are at (± 3, 0) and which passes through (4, 1).
Answer:
Page No 26.23:
Question 12:
Find the equation of an ellipse whose eccentricity is 2/3, the latus-rectum is 5 and the centre is at the origin.
Answer:
Page No 26.23:
Question 13:
Find the equation of an ellipse with its foci on y-axis, eccentricity 3/4, centre at the origin and passing through (6, 4).
Answer:
Page No 26.23:
Question 14:
Find the equation of an ellipse whose axes lie along coordinate axes and which passes through (4, 3) and (−1, 4).
Answer:
Page No 26.23:
Question 15:
Find the equation of an ellipse whose axes lie along the coordinate axes, which passes through the point (−3, 1) and has eccentricity equal to .
Answer:
Page No 26.23:
Question 16:
Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directrices is 18 units.
Answer:
Page No 26.23:
Question 17:
Find the equation of an ellipse whose vertices are (0, ± 10) and eccentricity e = .
Answer:
Page No 26.23:
Question 18:
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.
Answer:
Let AB be the rod making an angle θ with OX and let P (x, y) be the point on it such that AP = 3 cm.
Then, PB = AB – AP = (12 – 3) cm = 9 cm [∵ AB = 12 cm]
From P, draw PQ⊥OY and PR⊥OX.
Page No 26.23:
Question 19:
Find the equation of the set of all points whose distances from (0, 4) are of their distances from the line y = 9.
Answer:
We have
Page No 26.27:
Question 1:
If the lengths of semi-major and semi-minor axes of an ellipse are 2 and and their corresponding equations are y − 5 = 0 and x + 3 = 0, then write the equation of the ellipse.
Answer:
Page No 26.27:
Question 2:
Write the eccentricity of the ellipse 9x2 + 5y2 − 18x − 2y − 16 = 0.
Answer:
Page No 26.27:
Question 3:
Write the centre and eccentricity of the ellipse 3x2 + 4y2 − 6x + 8y − 5 = 0.
Answer:
Page No 26.27:
Question 4:
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.
Answer:
Page No 26.27:
Question 5:
Write the eccentricity of an ellipse whose latus-rectum is one half of the minor axis.
Answer:
Page No 26.27:
Question 6:
If the distance between the foci of an ellipse is equal to the length of the latus-rectum, write the eccentricity of the ellipse.
Answer:
Page No 26.27:
Question 7:
If S and S' are two foci of the ellipse and B is an end of the minor axis such that ∆BSS' is equilateral, then write the eccentricity of the ellipse.
Answer:
Page No 26.27:
Question 8:
If the minor axis of an ellipse subtends an equilateral triangle with vertex at one end of major axis, then write the eccentricity of the ellipse.
Answer:
Page No 26.27:
Question 9:
If a latus rectum of an ellipse subtends a right angle at the centre of the ellipse, then write the eccentricity of the ellipse.
Answer:
Page No 26.27:
Question 1:
For the ellipse 12x2 + 4y2 + 24x − 16y + 25 = 0
(a) centre is (−1, 2)
(b) lengths of the axes are and 1
(c) eccentricity =
(d) all of these
Answer:
Page No 26.28:
Question 2:
The equation of the ellipse with focus (−1, 1), directrix x − y + 3 = 0 and eccentricity 1/2 is
(a) 7x2 + 2xy + 7y2 + 10x + 10y + 7 = 0
(b) 7x2 + 2xy + 7y2 + 10x − 10y + 7 = 0
(c) 7x2 + 2xy + 7y2 + 10x − 10y − 7 = 0
(d) none of these
Answer:
Page No 26.28:
Question 3:
The equation of the circle drawn with the two foci of as the end-points of a diameter is
(a) x2 + y2 = a2 + b2
(b) x2 + y2 = a2
(c) x2 + y2 = 2a2
(d) x2 + y2 = a2 − b2
Answer:
Page No 26.28:
Question 4:
The eccentricity of the ellipse if its latus rectum is equal to one half of its minor axis, is
(a)
(b)
(c)
(d) none of these
Answer:
Page No 26.28:
Question 5:
The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus-rectum, is
(a)
(b)
(c)
(d) none of these
Answer:
Page No 26.28:
Question 6:
The eccentricity of the ellipse, if the minor axis is equal to the distance between the foci, is
(a)
(b)
(c)
(d)
Answer:
Page No 26.28:
Question 7:
The difference between the lengths of the major axis and the latus-rectum of an ellipse is
(a) ae
(b) 2ae
(c) ae2
(d) 2ae2
Answer:
Page No 26.28:
Question 8:
The eccentricity of the conic 9x2 + 25y2 = 225 is
(a) 2/5
(b) 4/5
(c) 1/3
(d) 1/5
(e) 3/5
Answer:
Page No 26.28:
Question 9:
The latus-rectum of the conic 3x2 + 4y2 − 6x + 8y − 5 = 0 is
(a) 3
(b)
(c)
(d) none of these
Answer:
Page No 26.28:
Question 10:
The equations of the tangents to the ellipse 9x2 + 16y2 = 144 from the point (2, 3) are
(a) y = 3, x = 5
(b) x = 2, y = 3
(c) x = 3, y = 2
(d) x + y = 5, y = 3
Answer:
Page No 26.28:
Question 11:
The eccentricity of the ellipse 4x2 + 9y2 + 8x + 36y + 4 = 0 is
(a)
(b)
(c)
(d)
Answer:
Page No 26.28:
Question 12:
The eccentricity of the ellipse 4x2 + 9y2 = 36 is
(a)
(b)
(c)
(d)
Answer:
Page No 26.28:
Question 13:
The eccentricity of the ellipse 5x2 + 9y2 = 1 is
(a) 2/3
(b) 3/4
(c) 4/5
(d) 1/2
Answer:
Page No 26.28:
Question 14:
For the ellipse x2 + 4y2 = 9
(a) the eccentricity is 1/2
(b) the latus-rectum is 3/2
(c) a focus is
(d) a directrix is x =
Answer:
Page No 26.29:
Question 15:
If the latus rectum of an ellipse is one half of its minor axis, then its eccentricity is
(a)
(b)
(c)
(d)
Answer:
Page No 26.29:
Question 16:
An ellipse has its centre at (1, −1) and semi-major axis = 8 and it passes through the point
(1, 3). The equation of the ellipse is
(a)
(b)
(c)
(d)
Answer:
Page No 26.29:
Question 17:
The sum of the focal distances of any point on the ellipse 9x2 + 16y2 = 144 is
(a) 32
(b) 18
(c) 16
(d) 8
Answer:
Page No 26.29:
Question 18:
If (2, 4) and (10, 10) are the ends of a latus-rectum of an ellipse with eccentricity 1/2, then the length of semi-major axis is
(a) 20/3
(b) 15/3
(c) 40/3
(d) none of these
Answer:
Page No 26.29:
Question 19:
The equation represents an ellipse, if
(a) λ < 5
(b) λ < 2
(c) 2 < λ < 5
(d) λ < 2 or λ > 5
Answer:
Page No 26.29:
Question 20:
The eccentricity of the ellipse 9x2 + 25y2 − 18x − 100y − 116 = 0, is
(a) 25/16
(b) 4/5
(c) 16/25
(d) 5/4
Answer:
Page No 26.29:
Question 21:
If the major axis of an ellipse is three times the minor axis, then its eccentricity is equal to
(a)
(b)
(c)
(d)
(e)
Answer:
Page No 26.29:
Question 22:
The eccentricity of the ellipse 25x2 + 16y2 = 400 is
(a) 3/5
(b) 1/3
(c) 2/5
(d) 1/5
Answer:
Page No 26.29:
Question 23:
The eccentricity of the ellipse 5x2 + 9y2 = 1 is
(a) 2/3
(b) 3/4
(c) 4/5
(d) 1/2
Answer:
Page No 26.29:
Question 24:
The eccentricity of the ellipse 4x2 + 9y2 = 36 is
(a)
(b)
(c)
(d)
Answer:
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