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Conic Sections
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NCERT Solutions
Conic Sections
70 Solutions
Exercise:
All Exercises
Exercise 10.1
Exercise 10.2
Exercise 10.3
Exercise 10.4
Miscellaneous Exercise on Chapter 10
Q1
Exercise 10.1
In each of the following Exercises 1 to 5, find the equation of the circle with
centre
(
0
,
2
)
(0,2)
(
0
,
2
)
and radius 2
Q2
Exercise 10.1
centre
(
−
2
,
3
)
(-2,3)
(
−
2
,
3
)
and radius 4
Q3
Exercise 10.1
centre
(
1
2
,
1
4
)
\left(\frac{1}{2}, \frac{1}{4}\right)
(
2
1
,
4
1
)
and radius
1
12
\frac{1}{12}
12
1
Q4
Exercise 10.1
centre
(
1
,
1
)
(1,1)
(
1
,
1
)
and radius
2
\sqrt{2}
2
Q5
Exercise 10.1
centre
(
−
a
,
−
b
)
(-a,-b)
(
−
a
,
−
b
)
and radius
a
2
−
b
2
\sqrt{a^{2}-b^{2}}
a
2
−
b
2
Q6
Exercise 10.1
In each of the following Exercises 6 to 9, find the centre and radius of the circles. 6.
(
x
+
5
)
2
+
(
y
−
3
)
2
=
36
(x+5)^{2}+(y-3)^{2}=36
(
x
+
5
)
2
+
(
y
−
3
)
2
=
36
Q7
Exercise 10.1
x
2
+
y
2
−
4
x
−
8
y
−
45
=
0
x^{2}+y^{2}-4 x-8 y-45=0
x
2
+
y
2
−
4
x
−
8
y
−
45
=
0
Q8
Exercise 10.1
x
2
+
y
2
−
8
x
+
10
y
−
12
=
0
x^{2}+y^{2}-8 x+10 y-12=0
x
2
+
y
2
−
8
x
+
10
y
−
12
=
0
Q9
Exercise 10.1
2
x
2
+
2
y
2
−
x
=
0
2 x^{2}+2 y^{2}-x=0
2
x
2
+
2
y
2
−
x
=
0
Q10
Exercise 10.1
Find the equation of the circle passing through the points
(
4
,
1
)
(4,1)
(
4
,
1
)
and
(
6
,
5
)
(6,5)
(
6
,
5
)
and whose centre is on the line
4
x
+
y
=
16
4 x+y=16
4
x
+
y
=
16
.
Q11
Exercise 10.1
Find the equation of the circle passing through the points
(
2
,
3
)
(2,3)
(
2
,
3
)
and
(
−
1
,
1
)
(-1,1)
(
−
1
,
1
)
and whose centre is on the line
x
−
3
y
−
11
=
0
x-3 y-11=0
x
−
3
y
−
11
=
0
.
Q12
Exercise 10.1
Find the equation of the circle with radius 5 whose centre lies on
x
x
x
-axis and passes through the point
(
2
,
3
)
(2,3)
(
2
,
3
)
.
Q13
Exercise 10.1
Find the equation of the circle passing through
(
0
,
0
)
(0,0)
(
0
,
0
)
and making intercepts
a
a
a
and
b
b
b
on the coordinate axes.
Q14
Exercise 10.1
Find the equation of a circle with centre
(
2
,
2
)
(2,2)
(
2
,
2
)
and passes through the point
(
4
,
5
)
(4,5)
(
4
,
5
)
.
Q15
Exercise 10.1
Does the point
(
−
2.5
,
3.5
)
(-2.5, 3.5)
(
−
2.5
,
3.5
)
lie inside, outside or on the circle
x
2
+
y
2
=
25
x^{2}+y^{2}=25
x
2
+
y
2
=
25
?
Q1
Exercise 10.2
In each of the following Exercises 1 to 6, find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
y
2
=
12
x
y^{2}=12 x
y
2
=
12
x
Q2
Exercise 10.2
x
2
=
6
y
x^{2}=6 y
x
2
=
6
y
Q3
Exercise 10.2
y
2
=
−
8
x
y^{2}=-8 x
y
2
=
−
8
x
Q4
Exercise 10.2
x
2
=
−
16
y
x^{2}=-16 y
x
2
=
−
16
y
Q5
Exercise 10.2
y
2
=
10
x
y^{2}=10 x
y
2
=
10
x
Q6
Exercise 10.2
x
2
=
−
9
y
x^{2}=-9 y
x
2
=
−
9
y
Q7
Exercise 10.2
In each of the Exercises 7 to 12, find the equation of the parabola that satisfies the given conditions: 7. Focus
(
6
,
0
)
(6,0)
(
6
,
0
)
; directrix
x
=
−
6
x=-6
x
=
−
6
Q8
Exercise 10.2
Focus
(
0
,
−
3
)
(0,-3)
(
0
,
−
3
)
; directrix
y
=
3
y=3
y
=
3
Q9
Exercise 10.2
Vertex
(
0
,
0
)
(0,0)
(
0
,
0
)
; focus
(
3
,
0
)
(3,0)
(
3
,
0
)
Q10
Exercise 10.2
Vertex
(
0
,
0
)
(0,0)
(
0
,
0
)
; focus
(
−
2
,
0
)
(-2,0)
(
−
2
,
0
)
Q11
Exercise 10.2
Vertex
(
0
,
0
)
(0,0)
(
0
,
0
)
passing through
(
2
,
3
)
(2,3)
(
2
,
3
)
and axis is along
x
x
x
-axis.
Q12
Exercise 10.2
Vertex
(
0
,
0
)
(0,0)
(
0
,
0
)
, passing through
(
5
,
2
)
(5,2)
(
5
,
2
)
and symmetric with respect to
y
y
y
-axis.
Q1
Exercise 10.3
In each of the Exercises 1 to 9, 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
36
+
y
2
16
=
1
\frac{x^{2}}{36}+\frac{y^{2}}{16}=1
36
x
2
+
16
y
2
=
1
Q2
Exercise 10.3
x
2
4
+
y
2
25
=
1
\frac{x^{2}}{4}+\frac{y^{2}}{25}=1
4
x
2
+
25
y
2
=
1
Q3
Exercise 10.3
x
2
16
+
y
2
9
=
1
\frac{x^{2}}{16}+\frac{y^{2}}{9}=1
16
x
2
+
9
y
2
=
1
Q4
Exercise 10.3
x
2
25
+
y
2
100
=
1
\frac{x^{2}}{25}+\frac{y^{2}}{100}=1
25
x
2
+
100
y
2
=
1
Q5
Exercise 10.3
x
2
49
+
y
2
36
=
1
\frac{x^{2}}{49}+\frac{y^{2}}{36}=1
49
x
2
+
36
y
2
=
1
Q6
Exercise 10.3
x
2
100
+
y
2
400
=
1
\frac{x^{2}}{100}+\frac{y^{2}}{400}=1
100
x
2
+
400
y
2
=
1
Q7
Exercise 10.3
36
x
2
+
4
y
2
=
144
36 x^{2}+4 y^{2}=144
36
x
2
+
4
y
2
=
144
Q8
Exercise 10.3
16
x
2
+
y
2
=
16
16 x^{2}+y^{2}=16
16
x
2
+
y
2
=
16
Q9
Exercise 10.3
4
x
2
+
9
y
2
=
36
4 x^{2}+9 y^{2}=36
4
x
2
+
9
y
2
=
36
Q10
Exercise 10.3
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions: 10. Vertices
(
±
5
,
0
)
( \pm 5,0)
(
±
5
,
0
)
, foci
(
±
4
,
0
)
( \pm 4,0)
(
±
4
,
0
)
Q11
Exercise 10.3
Vertices
(
0
,
±
13
)
(0, \pm 13)
(
0
,
±
13
)
, foci
(
0
,
±
5
)
(0, \pm 5)
(
0
,
±
5
)
Q12
Exercise 10.3
Vertices
(
±
6
,
0
)
( \pm 6,0)
(
±
6
,
0
)
, foci
(
±
4
,
0
)
( \pm 4,0)
(
±
4
,
0
)
Q13
Exercise 10.3
Ends of major axis
(
±
3
,
0
)
( \pm 3,0)
(
±
3
,
0
)
, ends of minor axis
(
0
,
±
2
)
(0, \pm 2)
(
0
,
±
2
)
Q14
Exercise 10.3
Ends of major axis
(
0
,
±
5
)
(0, \pm \sqrt{5})
(
0
,
±
5
)
, ends of minor axis
(
±
1
,
0
)
( \pm 1,0)
(
±
1
,
0
)
Q15
Exercise 10.3
Length of major axis 26, foci
(
±
5
,
0
)
( \pm 5,0)
(
±
5
,
0
)
Q16
Exercise 10.3
Length of minor axis 16, foci
(
0
,
±
6
)
(0, \pm 6)
(
0
,
±
6
)
.
Q17
Exercise 10.3
Foci
(
±
3
,
0
)
,
a
=
4
( \pm 3,0), a=4
(
±
3
,
0
)
,
a
=
4
Q18
Exercise 10.3
b
=
3
,
c
=
4
b=3, c=4
b
=
3
,
c
=
4
, centre at the origin; foci on the
x
x
x
axis.
Q19
Exercise 10.3
Centre at
(
0
,
0
)
(0,0)
(
0
,
0
)
, major axis on the
y
y
y
-axis and passes through the points
(
3
,
2
)
(3,2)
(
3
,
2
)
and
(
1
,
6
)
(1,6)
(
1
,
6
)
.
Q20
Exercise 10.3
Major axis on the
x
x
x
-axis and passes through the points
(
4
,
3
)
(4,3)
(
4
,
3
)
and
(
6
,
2
)
(6,2)
(
6
,
2
)
.
Q1
Exercise 10.4
In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.
x
2
16
−
y
2
9
=
1
\frac{x^{2}}{16}-\frac{y^{2}}{9}=1
16
x
2
−
9
y
2
=
1
Q2
Exercise 10.4
y
2
9
−
x
2
27
=
1
\frac{y^{2}}{9}-\frac{x^{2}}{27}=1
9
y
2
−
27
x
2
=
1
Q3
Exercise 10.4
9
y
2
−
4
x
2
=
36
9 y^{2}-4 x^{2}=36
9
y
2
−
4
x
2
=
36
Q4
Exercise 10.4
16
x
2
−
9
y
2
=
576
16 x^{2}-9 y^{2}=576
16
x
2
−
9
y
2
=
576
Q5
Exercise 10.4
5
y
2
−
9
x
2
=
36
5 y^{2}-9 x^{2}=36
5
y
2
−
9
x
2
=
36
Q6
Exercise 10.4
49
y
2
−
16
x
2
=
784
49 y^{2}-16 x^{2}=784
49
y
2
−
16
x
2
=
784
.
Q7
Exercise 10.4
In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions. 7. Vertices
(
±
2
,
0
)
( \pm 2,0)
(
±
2
,
0
)
, foci
(
±
3
,
0
)
( \pm 3,0)
(
±
3
,
0
)
Q8
Exercise 10.4
Vertices
(
0
,
±
5
)
(0, \pm 5)
(
0
,
±
5
)
, foci
(
0
,
±
8
)
(0, \pm 8)
(
0
,
±
8
)
Q9
Exercise 10.4
Vertices
(
0
,
±
3
)
(0, \pm 3)
(
0
,
±
3
)
, foci
(
0
,
±
5
)
(0, \pm 5)
(
0
,
±
5
)
Q10
Exercise 10.4
Foci
(
±
5
,
0
)
( \pm 5,0)
(
±
5
,
0
)
, the transverse axis is of length 8.
Q11
Exercise 10.4
Foci
(
0
,
±
13
)
(0, \pm 13)
(
0
,
±
13
)
, the conjugate axis is of length 24.
Q12
Exercise 10.4
Foci
(
±
3
5
,
0
)
( \pm 3 \sqrt{5}, 0)
(
±
3
5
,
0
)
, the latus rectum is of length 8.
Q13
Exercise 10.4
Foci
(
±
4
,
0
)
( \pm 4,0)
(
±
4
,
0
)
, the latus rectum is of length 12
Q14
Exercise 10.4
vertices
(
±
7
,
0
)
,
e
=
4
3
( \pm 7,0), e=\frac{4}{3}
(
±
7
,
0
)
,
e
=
3
4
.
Q15
Exercise 10.4
Foci
(
0
,
±
10
)
(0, \pm \sqrt{10})
(
0
,
±
10
)
, passing through
(
2
,
3
)
(2,3)
(
2
,
3
)
Q1
Miscellaneous Exercise on Chapter 10
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
Q2
Miscellaneous Exercise on Chapter 10
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
Q3
Miscellaneous Exercise on Chapter 10
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
Q4
Miscellaneous Exercise on Chapter 10
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.
Q5
Miscellaneous Exercise on Chapter 10
A rod of length 12 cm 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 the
x
x
x
-axis.
Q6
Miscellaneous Exercise on Chapter 10
Find the area of the triangle formed by the lines joining the vertex of the parabola
x
2
=
12
y
x^{2}=12 y
x
2
=
12
y
to the ends of its latus rectum.
Q7
Miscellaneous Exercise on Chapter 10
A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.
Q8
Miscellaneous Exercise on Chapter 10
An equilateral triangle is inscribed in the parabola
y
2
=
4
a
x
y^{2}=4 a x
y
2
=
4
a
x
, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
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