Mathematics · Textbook solutions

Conic Sections

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 90 questions

10.3 Circle

19 q

Solved Examples

Worked · 4
  1. 10.1 Eg.1
    Find an equation of the circle with centre at (0,0)(0,0) and radius rr.
  2. 10.1 Eg.2
    Find the equation of the circle with centre (−3,2)(-3, 2) and radius 4.
  3. 10.1 Eg.3
    Find the centre and the radius of the circle x2+y2+8x+10y−8=0x^2 + y^2 + 8x + 10y - 8 = 0.
  4. 10.1 Eg.4
    Find the equation of the circle which passes through the points (2,−2)(2, -2), and (3,4)(3, 4) and whose centre lies on the line x+y=2x + y = 2.

Exercise 10.1

Practice · 15
  1. Ex 10.1 Q1
    Find the equation of the circle with centre (0,2)(0,2) and radius 2.
  2. Ex 10.1 Q2
    Find the equation of the circle with centre (−2,3)(-2,3) and radius 4.
  3. Ex 10.1 Q3
    Find the equation of the circle with centre (12,14)\left(\frac{1}{2}, \frac{1}{4}\right) and radius 112\frac{1}{12}.
  4. Ex 10.1 Q4
    Find the equation of the circle with centre (1,1)(1,1) and radius 2\sqrt{2}.
  5. Ex 10.1 Q5
    Find the equation of the circle with centre (−a,−b)(-a, -b) and radius a2−b2\sqrt{a^2 - b^2}.
  6. Ex 10.1 Q6
    Find the centre and radius of the circle (x+5)2+(y−3)2=36(x + 5)^2 + (y - 3)^2 = 36.
  7. Ex 10.1 Q7
    Find the centre and radius of the circle x2+y2−4x−8y−45=0x^2 + y^2 - 4x - 8y - 45 = 0.
  8. Ex 10.1 Q8
    Find the centre and radius of the circle x2+y2−8x+10y−12=0x^2 + y^2 - 8x + 10y - 12 = 0.
  9. Ex 10.1 Q9
    Find the centre and radius of the circle 2x2+2y2−x=02x^2 + 2y^2 - x = 0.
  10. Ex 10.1 Q10
    Find the equation of the circle passing through the points (4,1)(4,1) and (6,5)(6,5) and whose centre is on the line 4x+y=164x + y = 16.
  11. Ex 10.1 Q11
    Find the equation of the circle passing through the points (2,3)(2,3) and (−1,1)(-1,1) and whose centre is on the line x−3y−11=0x - 3y - 11 = 0.
  12. Ex 10.1 Q12
    Find the equation of the circle with radius 5 whose centre lies on xx-axis and passes through the point (2,3)(2,3).
  13. Ex 10.1 Q13
    Find the equation of the circle passing through (0,0)(0,0) and making intercepts aa and bb on the coordinate axes.
  14. Ex 10.1 Q14
    Find the equation of a circle with centre (2,2)(2,2) and passes through the point (4,5)(4,5).
  15. Ex 10.1 Q15
    Does the point (−2.5,3.5)(-2.5, 3.5) lie inside, outside or on the circle x2+y2=25x^2 + y^2 = 25?

10.4 Parabola

16 q

Solved Examples

Worked · 4
  1. 10.2 Eg.5
    Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola y2=8xy^2 = 8x.
  2. 10.2 Eg.6
    Find the equation of the parabola with focus (2,0)(2, 0) and directrix x=−2x = -2.
  3. 10.2 Eg.7
    Find the equation of the parabola with vertex at (0,0)(0, 0) and focus at (0,2)(0, 2).
  4. 10.2 Eg.8
    Find the equation of the parabola which is symmetric about the yy-axis, and passes through the point (2,−3)(2, -3).

Exercise 10.2

Practice · 12
  1. Ex 10.2 Q1
    Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of y2=12xy^2 = 12x.
  2. Ex 10.2 Q2
    Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of x2=6yx^2 = 6y.
  3. Ex 10.2 Q3
    Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of y2=−8xy^2 = -8x.
  4. Ex 10.2 Q4
    Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of x2=−16yx^2 = -16y.
  5. Ex 10.2 Q5
    Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of y2=10xy^2 = 10x.
  6. Ex 10.2 Q6
    Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of x2=−9yx^2 = -9y.
  7. Ex 10.2 Q7
    Find the equation of the parabola that satisfies the given conditions: Focus (6,0)(6, 0); directrix x=−6x = -6.
  8. Ex 10.2 Q8
    Find the equation of the parabola that satisfies the given conditions: Focus (0,−3)(0, -3); directrix y=3y = 3.
  9. Ex 10.2 Q9
    Find the equation of the parabola that satisfies the given conditions: Vertex (0,0)(0, 0); focus (3,0)(3, 0).
  10. Ex 10.2 Q10
    Find the equation of the parabola that satisfies the given conditions: Vertex (0,0)(0, 0); focus (−2,0)(-2, 0).
  11. Ex 10.2 Q11
    Find the equation of the parabola that satisfies the given conditions: Vertex (0,0)(0, 0) passing through (2,3)(2, 3) and axis is along xx-axis.
  12. Ex 10.2 Q12
    Find the equation of the parabola that satisfies the given conditions: Vertex (0,0)(0, 0), passing through (5,2)(5, 2) and symmetric with respect to yy-axis.

10.5 Ellipse

25 q

Solved Examples

Worked · 5
  1. 10.3 Eg.9
    Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the latus rectum of the ellipse x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1.
  2. 10.3 Eg.10
    Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse 9x2+4y2=369x^2 + 4y^2 = 36.
  3. 10.3 Eg.11
    Find the equation of the ellipse whose vertices are (±13,0)(\pm 13, 0) and foci are (±5,0)(\pm 5, 0).
  4. 10.3 Eg.12
    Find the equation of the ellipse, whose length of the major axis is 20 and foci are (0,±5)(0, \pm 5).
  5. 10.3 Eg.13
    Find the equation of the ellipse, with major axis along the xx-axis and passing through the points (4,3)(4, 3) and (−1,4)(-1, 4).

Exercise 10.3

Practice · 20
  1. Ex 10.3 Q1
    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 x236+y216=1\frac{x^2}{36} + \frac{y^2}{16} = 1.
  2. Ex 10.3 Q2
    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 x24+y225=1\frac{x^2}{4} + \frac{y^2}{25} = 1.
  3. Ex 10.3 Q3
    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 x216+y29=1\frac{x^2}{16} + \frac{y^2}{9} = 1.
  4. Ex 10.3 Q4
    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 x225+y2100=1\frac{x^2}{25} + \frac{y^2}{100} = 1.
  5. Ex 10.3 Q5
    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 x249+y236=1\frac{x^2}{49} + \frac{y^2}{36} = 1.
  6. Ex 10.3 Q6
    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 x2100+y2400=1\frac{x^2}{100} + \frac{y^2}{400} = 1.
  7. Ex 10.3 Q7
    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=14436x^2 + 4y^2 = 144.
  8. Ex 10.3 Q8
    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=1616x^2 + y^2 = 16.
  9. Ex 10.3 Q9
    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=364x^2 + 9y^2 = 36.
  10. Ex 10.3 Q10
    Find the equation for the ellipse that satisfies the given conditions: Vertices (±5,0)(\pm 5, 0), foci (±4,0)(\pm 4, 0).
  11. Ex 10.3 Q11
    Find the equation for the ellipse that satisfies the given conditions: Vertices (0,±13)(0, \pm 13), foci (0,±5)(0, \pm 5).
  12. Ex 10.3 Q12
    Find the equation for the ellipse that satisfies the given conditions: Vertices (±6,0)(\pm 6, 0), foci (±4,0)(\pm 4, 0).
  13. Ex 10.3 Q13
    Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3,0)(\pm 3, 0), ends of minor axis (0,±2)(0, \pm 2).
  14. Ex 10.3 Q14
    Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0,±5)(0, \pm \sqrt{5}), ends of minor axis (±1,0)(\pm 1, 0).
  15. Ex 10.3 Q15
    Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5,0)(\pm 5, 0).
  16. Ex 10.3 Q16
    Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0,±6)(0, \pm 6).
  17. Ex 10.3 Q17
    Find the equation for the ellipse that satisfies the given conditions: Foci (±3,0)(\pm 3, 0), a=4a = 4.
  18. Ex 10.3 Q18
    Find the equation for the ellipse that satisfies the given conditions: b=3b = 3, c=4c = 4, centre at the origin; foci on the xx axis.
  19. Ex 10.3 Q19
    Find the equation for the ellipse that satisfies the given conditions: Centre at (0,0)(0, 0), major axis on the yy-axis and passes through the points (3,2)(3, 2) and (1,6)(1, 6).
  20. Ex 10.3 Q20
    Find the equation for the ellipse that satisfies the given conditions: Major axis on the xx-axis and passes through the points (4,3)(4, 3) and (6,2)(6, 2).

10.6 Hyperbola

19 q

Solved Examples

Worked · 4
  1. Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbolas:
    10.4 Eg.14(i)
    x29−y216=1\frac{x^2}{9} - \frac{y^2}{16} = 1
  2. 10.4 Eg.14(ii)
    y2−16x2=16y^2 - 16x^2 = 16
  3. 10.4 Eg.15
    Find the equation of the hyperbola with foci (0,±3)(0, \pm 3) and vertices (0,±112)\left(0, \pm\frac{\sqrt{11}}{2}\right).
  4. 10.4 Eg.16
    Find the equation of the hyperbola where foci are (0,±12)(0, \pm 12) and the length of the latus rectum is 36.

Exercise 10.4

Practice · 15
  1. Ex 10.4 Q1
    Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola x216−y29=1\frac{x^2}{16} - \frac{y^2}{9} = 1.
  2. Ex 10.4 Q2
    Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola y29−x227=1\frac{y^2}{9} - \frac{x^2}{27} = 1.
  3. Ex 10.4 Q3
    Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola 9y2−4x2=369y^2 - 4x^2 = 36.
  4. Ex 10.4 Q4
    Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola 16x2−9y2=57616x^2 - 9y^2 = 576.
  5. Ex 10.4 Q5
    Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola 5y2−9x2=365y^2 - 9x^2 = 36.
  6. Ex 10.4 Q6
    Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola 49y2−16x2=78449y^2 - 16x^2 = 784.
  7. Ex 10.4 Q7
    Find the equation of the hyperbola satisfying the given conditions: Vertices (±2,0)(\pm 2, 0), foci (±3,0)(\pm 3, 0).
  8. Ex 10.4 Q8
    Find the equation of the hyperbola satisfying the given conditions: Vertices (0,±5)(0, \pm 5), foci (0,±8)(0, \pm 8).
  9. Ex 10.4 Q9
    Find the equation of the hyperbola satisfying the given conditions: Vertices (0,±3)(0, \pm 3), foci (0,±5)(0, \pm 5).
  10. Ex 10.4 Q10
    Find the equation of the hyperbola satisfying the given conditions: Foci (±5,0)(\pm 5, 0), the transverse axis is of length 8.
  11. Ex 10.4 Q11
    Find the equation of the hyperbola satisfying the given conditions: Foci (0,±13)(0, \pm 13), the conjugate axis is of length 24.
  12. Ex 10.4 Q12
    Find the equation of the hyperbola satisfying the given conditions: Foci (±35,0)(\pm 3\sqrt{5}, 0), the latus rectum is of length 8.
  13. Ex 10.4 Q13
    Find the equation of the hyperbola satisfying the given conditions: Foci (±4,0)(\pm 4, 0), the latus rectum is of length 12.
  14. Ex 10.4 Q14
    Find the equation of the hyperbola satisfying the given conditions: vertices (±7,0)(\pm 7, 0), e=43e = \frac{4}{3}.
  15. Ex 10.4 Q15
    Find the equation of the hyperbola satisfying the given conditions: Foci (0,±10)(0, \pm\sqrt{10}), passing through (2,3)(2, 3).

Miscellaneous Exercise on Chapter 10

11 q

Miscellaneous Examples

Worked · 3
  1. Misc Eg.17
    The focus of a parabolic mirror as shown in Fig 10.31 is at a distance of 5 cm from its vertex. If the mirror is 45 cm deep, find the distance AB (Fig 10.31).
  2. Misc Eg.18
    A beam is supported at its ends by supports which are 12 metres apart. Since the load is concentrated at its centre, there is a deflection of 3 cm at the centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection 1 cm?
  3. Misc Eg.19
    A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on xx-axis and end point B lies on yy-axis. A point P(x,y)(x, y) is taken on the rod in such a way that AP = 6 cm. Show that the locus of P is an ellipse.

Miscellaneous Exercise

Practice · 8
  1. Misc Q1
    If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
  2. Misc Q2
    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?
  3. Misc Q3
    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.
  4. Misc Q4
    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.
  5. Misc Q5
    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 xx-axis.
  6. Misc Q6
    Find the area of the triangle formed by the lines joining the vertex of the parabola x2=12yx^2 = 12y to the ends of its latus rectum.
  7. Misc Q7
    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.
  8. Misc Q8
    An equilateral triangle is inscribed in the parabola y2=4axy^2 = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.