Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution

Question 1.
Find the
(i) lengths of the principal axes
(ii) co-ordinates of the foci
(iii) equations of directrices
(iv) length of the latus rectum
(v) distance between foci
(vi) distance between directrices of the ellipse:
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a = 5 and b = 3
Since a > b,
X-axis is the major axis and Y-axis is the minor axis.

(i) Length of major axis = 2a = 2(5) = 10
Length of minor axis = 2b = 2(3) = 6
Lengths of the principal axes are 10 and 6.

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(i) Length of major axis = 2a = 2(2) = 4
Length of minor axis = 2b = 2√3
Lengths of the principal axes are 4 and 2√3.

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a = √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 = 2√3
Length of minor axis = 2b = 2(1) = 2
Lengths of the principal axes are 2√3 and 2.

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Question 2.
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Distance between foci = 2ae
Given, distance between foci = 8
2ae = 8
2(5)e = 8
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Maharashtra Board 11th Maths Solutions Chapter 7 Conic Sections Ex 7.2 Q2 (iv)
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(vi) Given, the length of the latus rectum is 6, and co-ordinates of foci are (±2, 0).
The foci of the ellipse are on the X-axis.
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The ellipse passes through the points (-3, 1) and (2, -2).
Substituting x = -3 and y = 1 in equation of ellipse, we get
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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
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The ellipse passes through (-√5, 2).
Substituting x = -√5 and y = 2 in equation of ellipse, we get
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Question 3.
Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.
Solution:

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Question 4.
Find the eccentricity of an ellipse, if the distance between its directrices is three times the distance between its foci.
Solution:

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Question 5.
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Question 6.
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Question 7.
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Question 8.
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Question 9.
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Question 10.
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Question 11.

Solution:

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Slope of the given line x + y + 1 = 0 is -1.
Since the given line is parallel to the required tangents,
the slope of the required tangents is m = -1.
Equations of tangents to the ellipse
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Maharashtra Board 11th Maths Solutions Chapter 7 Conic Sections Ex 7.2 Q11 (vii)

Question 12.

Alternate method:
The locus of the point of intersection of perpendicular tangents is the director circle of an ellipse.
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Question 13.
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Question 14.
Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse.
Solution:

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Question 15.
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Question 16.
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Question 17.
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Question 18.
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Maharashtra Board 11th Maths Solutions Chapter 7 Conic Sections Ex 7.2 Q18
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