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Solutions

Overview

Step-by-step NCERT solutions for Conic Sections (Chapter 10, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Conic Sections textbook chapter.

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What these solutions cover

All 70 questions in Conic Sections are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 10.1

  1. Find the equation of the circle with centre (0, 2) and radius 2.
  2. Find the equation of the circle with centre (–2, 3) and radius 4.
  3. Find the equation of the circle with centre (1/2, 1/4) and radius 1/12.
  4. Find the equation of the circle with centre (1, 1) and radius √2.
  5. Find the equation of the circle with centre (–a, –b) and radius √(a² – b²).
  6. Find the centre and radius of the circle (x + 5)² + (y – 3)² = 36.
  7. Find the centre and radius of the circle x² + y² – 4x – 8y – 45 = 0.
  8. Find the centre and radius of the circle x² + y² – 8x + 10y – 12 = 0.
  9. Find the centre and radius of the circle 2x² + 2y² – x = 0.
  10. Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.
  11. Find the equation of the circle passing through the points (2, 3) and (–1, 1) and whose centre is on the line x – 3y – 11 = 0.
  12. Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2, 3).
  13. Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.
  14. Find the equation of a circle with centre (2, 2) and passes through the point (4, 5).
  15. Does the point (–2.5, 3.5) lie inside, outside or on the circle x² + y² = 25?

Exercise 10.2

  1. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: y² = 12x.
  2. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: x² = 6y.
  3. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: y² = –8x.
  4. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: x² = –16y.
  5. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: y² = 10x.
  6. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: x² = –9y.
  7. Find the equation of the parabola that satisfies the given conditions: Focus (6, 0); directrix x = –6.
  8. Find the equation of the parabola that satisfies the given conditions: Focus (0, –3); directrix y = 3.
  9. Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0); focus (3, 0).
  10. Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0); focus (–2, 0).
  11. Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0) passing through (2, 3) and axis is along x-axis.
  12. Find the equation of the parabola that satisfies the given conditions: Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis.

Exercise 10.3

  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: x²/36 + y²/16 = 1.
  2. 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: x²/4 + y²/25 = 1.
  3. 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: x²/16 + y²/9 = 1.
  4. 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: x²/25 + y²/100 = 1.
  5. 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: x²/49 + y²/36 = 1.
  6. 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: x²/100 + y²/400 = 1.
  7. 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.
  8. 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.
  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: 4x² + 9y² = 36.
  10. Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0).
  11. Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5).
  12. Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0).
  13. Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3, 0), ends of minor axis (0, ±2).
  14. Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0, ±√5), ends of minor axis (±1, 0).
  15. Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0).
  16. Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6).
  17. Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), a = 4.
  18. Find the equation for the ellipse that satisfies the given conditions: b = 3, c = 4, centre at the origin; foci on the x-axis.
  19. Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
  20. Find the equation for the ellipse that satisfies the given conditions: Major axis on the x-axis and passes through the points (4, 3) and (6, 2).

Exercise 10.4

  1. Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola: x²/16 – y²/9 = 1.
  2. Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola: y²/9 – x²/27 = 1.
  3. Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola: 9y² – 4x² = 36.
  4. Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola: 16x² – 9y² = 576.
  5. Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola: 5y² – 9x² = 36.
  6. Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola: 49y² – 16x² = 784.
  7. Find the equations of the hyperbola satisfying the given conditions: Vertices (±2, 0), foci (±3, 0).
  8. Find the equations of the hyperbola satisfying the given conditions: Vertices (0, ±5), foci (0, ±8).
  9. Find the equations of the hyperbola satisfying the given conditions: Vertices (0, ±3), foci (0, ±5).
  10. Find the equations of the hyperbola satisfying the given conditions: Foci (±5, 0), the transverse axis is of length 8.
  11. Find the equations of the hyperbola satisfying the given conditions: Foci (0, ±13), the conjugate axis is of length 24.
  12. Find the equations of the hyperbola satisfying the given conditions: Foci (±3√5, 0), the latus rectum is of length 8.
  13. Find the equations of the hyperbola satisfying the given conditions: Foci (±4, 0), the latus rectum is of length 12.
  14. Find the equations of the hyperbola satisfying the given conditions: Vertices (±7, 0), e = 4/3.
  15. Find the equations of the hyperbola satisfying the given conditions: Foci (0, ±√10), passing through (2, 3).

Miscellaneous Exercise

  1. If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
  2. 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. 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. 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. 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-axis.
  6. Find the area of the triangle formed by the lines joining the vertex of the parabola x² = 12y to the ends of its latus rectum.
  7. 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 path traced by the man.
  8. An equilateral triangle is inscribed in the parabola y² = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
Sample

Worked examples

3 questions from Conic Sections, solved step by step.

Exercise 10.1

Find the equation of the circle with centre (0, 2) and radius 2.

The standard equation of a circle with centre (h, k) and radius r is (x - h)² + (y - k)² = r².

Here h = 0, k = 2, r = 2.

Substituting: (x - 0)² + (y - 2)² = 4

Answer: x² + (y - 2)² = 4

Exercise 10.2

Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: y² = 12x.

The equation y² = 12x is of the form y² = 4ax, where 4a = 12, so a = 3.

Since the equation involves y² with positive coefficient of x, the parabola opens to the right.

Focus: (a, 0) = (3, 0).

Axis: x-axis (y = 0).

Directrix: x = -a ⇒ x = -3.

Latus rectum: 4a = 12.

Answer: Focus (3, 0); axis: x-axis; directrix x = –3; length of latus rectum = 12

Exercise 10.3

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: x²/36 + y²/16 = 1.

Here a² = 36, b² = 16, so a = 6, b = 4. Since a > b, the major axis is along the x-axis.

c² = a² - b² = 36 - 16 = 20 ⇒ c = 2√5.

Foci: (±c, 0) = (±2√5, 0).

Vertices: (±a, 0) = (±6, 0).

Major axis length: 2a = 12.

Minor axis length: 2b = 8.

Eccentricity: e = c/a = 2√5/6 = √5/3.

Latus rectum: 2b²/a = 2(16)/6 = 32/6 = 16/3.

Answer: Foci (±2√5, 0); vertices (±6, 0); major axis 12; minor axis 8; e = √5/3; latus rectum = 16/3

All 70 questions in Conic Sections are solved step by step in the Solutions PDF above.

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