Tamil Nadu Board of Secondary EducationHSC Science Class 12

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 5 - Two Dimensional Analytical Geometry-II [Latest edition]

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Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 5 - Two Dimensional Analytical Geometry-II - Shaalaa.com
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Solutions for Chapter 5: Two Dimensional Analytical Geometry-II

Below listed, you can find solutions for Chapter 5 of Tamil Nadu Board of Secondary Education Tamil Nadu Board Samacheer Kalvi for Class 12th Mathematics Volume 1 and 2 Answers Guide.


Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6
Exercise 5.1 [Page 182]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 5 Two Dimensional Analytical Geometry-II Exercise 5.1 [Page 182]

Exercise 5.1 | Q 1 | Page 182

Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form

Exercise 5.1 | Q 2 | Page 182

Find the equation of the circle with centre (2, −1) and passing through the point (3, 6) in standard form

Exercise 5.1 | Q 3 | Page 182

Find the equation of circles that touch both the axes and pass through (− 4, −2) in general form

Exercise 5.1 | Q 4 | Page 182

Find the equation of the circles with centre (2, 3) and passing through the intersection of the lines 3x – 2y – 1 = 0 and 4x + y – 27 = 0

Exercise 5.1 | Q 5 | Page 182

Obtain the equation of the circle for which (3, 4) and (2, -7) are the ends of a diameter

Exercise 5.1 | Q 6 | Page 182

Find the equation of the circle through the points (1, 0), (– 1, 0) and (0, 1)

Exercise 5.1 | Q 7 | Page 182

A circle of area 9π square units has two of its diameters along the lines x + y = 5 and x – y = 1. Find the equation of the circle

Exercise 5.1 | Q 8 | Page 182

If y = `2sqrt(2)x + "c"` is a tangent to the circle x2 + y2 = 16, find the value of c

Exercise 5.1 | Q 9 | Page 182

Find the equation of the tangent and normal to the circle x2 + y2 – 6x + 6y – 8 = 0 at (2, 2)

Exercise 5.1 | Q 10 | Page 182

Determine whether the points (– 2, 1), (0, 0) and (– 4, – 3) lie outside, on or inside the circle x2 + y2 – 5x + 2y – 5 = 0

Exercise 5.1 | Q 11. (i) | Page 182

Find centre and radius of the following circles

x2 + (y + 2)2 = 0

Exercise 5.1 | Q 11. (ii) | Page 182

Find centre and radius of the following circles

x2 + y2 + 6x – 4y + 4 = 0

Exercise 5.1 | Q 11. (iii) | Page 182

Find centre and radius of the following circles

x2 + y2 – x + 2y – 3 = 0

Exercise 5.1 | Q 11. (iv) | Page 182

Find centre and radius of the following circles

2x2 + 2y2 – 6x + 4y + 2 = 0

Exercise 5.1 | Q 12 | Page 182

If the equation 3x2 + (3 – p)xy + qy2 – 2px = 8pq represents a circle, find p and q. Also determine the centre and radius of the circle

Exercise 5.2 [Pages 196 - 197]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 5 Two Dimensional Analytical Geometry-II Exercise 5.2 [Pages 196 - 197]

Exercise 5.2 | Q 1. (i) | Page 196

Find the equation of the parabola in the cases given below:

Focus (4, 0) and directrix x = – 4

Exercise 5.2 | Q 1. (ii) | Page 196

Find the equation of the parabola in the cases given below:

Passes through (2, – 3) and symmetric about y-axis

Exercise 5.2 | Q 1. (iii) | Page 196

Find the equation of the parabola in the cases given below:

Vertex (1, – 2) and Focus (4, – 2)

Exercise 5.2 | Q 1. (iv) | Page 196

Find the equation of the parabola in the cases given below:

End points of latus rectum (4, – 8) and (4, 8)

Exercise 5.2 | Q 2. (i) | Page 196

Find the equation of the ellipse in the cases given below:

Foci `(+- 3, 0), "e"+ 1/2`

Exercise 5.2 | Q 2. (ii) | Page 196

Find the equation of the ellipse in the cases given below:

Foci (0, ±4) and end points of major axis are (0, ±5)

Exercise 5.2 | Q 2. (iii) | Page 196

Find the equation of the ellipse in the cases given below:

Length of latus rectum 8, eccentricity = `3/5` centre (0, 0) and major axis on x-axis

Exercise 5.2 | Q 2. (iv) | Page 196

Find the equation of the ellipse in the cases given below:

Length of latus rectum 4, distance between foci `4sqrt(2)`, centre (0, 0) and major axis as y-axis

Exercise 5.2 | Q 3. (i) | Page 196

Find the equation of the hyperbola in the cases given below:

Foci (± 2, 0), Eccentricity = `3/2`

Exercise 5.2 | Q 3. (ii) | Page 196

Find the equation of the hyperbola in the cases given below:

Centre (2, 1), one of the foci (8, 1) and corresponding directrix x = 4

Exercise 5.2 | Q 3. (iii) | Page 196

Find the equation of the hyperbola in the cases given below:

Passing through (5, – 2) and length of the transverse axis along x-axis and of length 8 units

Exercise 5.2 | Q 4. (i) | Page 197

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:

y2 = 16x

Exercise 5.2 | Q 4. (ii) | Page 197

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:

x2 = 24y

Exercise 5.2 | Q 4. (iii) | Page 197

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:

y2 = – 8x

Exercise 5.2 | Q 4. (iv) | Page 197

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:

x2 – 2x + 8y + 17 = 0

Exercise 5.2 | Q 4. (v) | Page 197

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:

y2 – 4y – 8x + 12 = 0

Exercise 5.2 | Q 5. (i) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`x^2/25 + y^2/9` = 1

Exercise 5.2 | Q 5. (ii) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`x^2/3 + y^2/10` = 1

Exercise 5.2 | Q 5. (iii) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`x^2/25 - y^2/144` = 1

Exercise 5.2 | Q 5. (iv) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`y^2/16 - x^2/9` = 1

Exercise 5.2 | Q 6 | Page 197

Prove that the length of the latus rectum of the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 is `(2"b"^2)/"a"`

Exercise 5.2 | Q 7 | Page 197

Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis

Exercise 5.2 | Q 8. (i) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`(x - 3)^2/225 + (y - 4)^2/289` = 1

Exercise 5.2 | Q 8. (ii) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`(x + 1)^2/100 + (y - 2)^2/64` = 1

Exercise 5.2 | Q 8. (iii) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`(x + 3)^2/225 + (y - 4)^2/64` = 1

Exercise 5.2 | Q 8. (iv) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`(y - 2)^3/25 + (x + 1)^2/16` = 1

Exercise 5.2 | Q 8. (v) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

18x2 + 12y2 – 144x + 48y + 120 = 0

Exercise 5.2 | Q 8. (vi) | Page 197

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

9x2 – y2 – 36x – 6y + 18 = 0

Exercise 5.3 [Page 199]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 5 Two Dimensional Analytical Geometry-II Exercise 5.3 [Page 199]

Exercise 5.3 | Q 1 | Page 199

Identify the type of conic section for the equation.

2x2 – y2 = 7

Exercise 5.3 | Q 2 | Page 199

Identify the type of conic section for the equation.

3x2 + 3y2 – 4x + 3y + 10 = 0

Exercise 5.3 | Q 3 | Page 199

Identify the type of conic section for the equation.

3x2 + 2y2 = 14

Exercise 5.3 | Q 4 | Page 199

Identify the type of conic section for the equation.

x2 + y2 + x – y = 0

Exercise 5.3 | Q 5 | Page 199

Identify the type of conic section for the equation.

11x2 – 25y2 – 44x + 50y – 256 = 0

Exercise 5.3 | Q 6 | Page 199

Identify the type of conic section for the equation.

y2 + 4x + 3y + 4 = 0

Exercise 5.4 [Pages 206 - 207]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 5 Two Dimensional Analytical Geometry-II Exercise 5.4 [Pages 206 - 207]

Exercise 5.4 | Q 1 | Page 206

Find the equations of the two tangents that can be drawn from (5, 2) to the ellipse 2x2 + 7y2 = 14

Exercise 5.4 | Q 2 | Page 206

Find the equations of tangents to the hyperbola `x^2/16 - y^2/64` = 1 which are parallel to10x − 3y + 9 = 0

Exercise 5.4 | Q 3 | Page 206

Show that the line x – y + 4 = 0 is a tangent to the ellipse x2 + 3y2 = 12. Also find the coordinates of the point of contact

Exercise 5.4 | Q 4 | Page 207

Find the equation of the tangent to the parabola y2 = 16x perpendicular to 2x + 2y + 3 = 0

Exercise 5.4 | Q 5 | Page 207

Find the equation of the tangent at t = 2 to the parabola y2 = 8x (Hint: use parametric form)

Exercise 5.4 | Q 6 | Page 207

Find the equations of the tangent and normal to hyperbola 12x2 – 9y2 = 108 at θ = `pi/3`. (Hint: use parametric form)

Exercise 5.4 | Q 7 | Page 207

Prove that the point of intersection of the tangents at ‘t1‘ and t2’ on the parabola y2 = 4ax is [at1 t2, a (t1 + t2)]

Exercise 5.4 | Q 8 | Page 207

If the normal at the point ‘t1‘ on the parabola y2 = 4ax meets the parabola again at the point ‘t2‘, then prove that t2 = `- ("t"_1 + 2/"t"_1)`

Exercise 5.5 [Pages 214 - 215]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 5 Two Dimensional Analytical Geometry-II Exercise 5.5 [Pages 214 - 215]

Exercise 5.5 | Q 1 | Page 214

A bridge has a parabolic arch that is 10 m high in the centre and 30 m wide at the bottom. Find the height of the arch 6m from the centre, on either sides

Exercise 5.5 | Q 2 | Page 214

A tunnel through a mountain for a four-lane highway is to have a elliptical opening. The total width of the highway (not the opening) is to be 16 m, and the height at the edge of the road must be sufficient for a truck 4 m high to clear if the highest point of the opening is to be 5 m approximately. How wide must the opening be?

Exercise 5.5 | Q 3 | Page 214

At a water fountain, water attains a maximum height of 4 m at horizontal distance of 0.5 m from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of 0.75 m from the point of origin.

Exercise 5.5 | Q 4. (a) | Page 214

An engineer designs a satellite dish with a parabolic cross-section. The dish is  5m wide at the opening, and the focus is placed 1 2. m from the vertex. Position a coordinate system with the origin at the vertex and the x-axis on the parabola’s axis of symmetry and find an equation of the parabola

Exercise 5.5 | Q 4. (b) | Page 214

An engineer designs a satellite dish with a parabolic cross-section. The dish is 5 m wide at the opening and the focus is placed 1.2 m from the vertex. Find the depth of the satellite dish at the vertex

Exercise 5.5 | Q 5 | Page 214

Parabolic cable of a 60 m portion of the roadbed of a suspension bridge are positioned as shown below. Vertical Cables are to be spaced every 6m along this portion of the roadbed. Calculate the lengths of first two of these vertical cables from the vertex.

Exercise 5.5 | Q 6 | Page 214

Cross-section of a Nuclear cooling tower is in the shape of a hyperbola with equation `x^2/30^2 - y^2/44^2` = 1. The tower is 150 m tall and the distance from the top of the tower to the centre of the hyperbola is half the distance from the base of the tower to the centre of the hyperbola. Find the diameter of the top and base of the tower

Exercise 5.5 | Q 7 | Page 215

A rod of length 1 2. m moves with its ends always touching the coordinate axes. The locus of a point P on the rod, which is 0 3. m from the end in contact with x-axis is an ellipse. Find the eccentricity

Exercise 5.5 | Q 8 | Page 215

Assume that water issuing from the end of a horizontal pipe, 7 5. m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2 5. m below the line of the pipe, the flow of water has curved outward 3 m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?

Exercise 5.5 | Q 9 | Page 215

On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4 m when it is 6m away from the point of projection. Finally it reaches the ground 12 m away from the starting point. Find the angle of projection

Exercise 5.5 | Q 10 | Page 215

Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.

Exercise 5.6 [Pages 215 - 217]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 5 Two Dimensional Analytical Geometry-II Exercise 5.6 [Pages 215 - 217]

Exercise 5.6 | Q 1 | Page 215

Choose the correct alternative:

The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis `x^2 + y^2 - 5x - 6y + 9 + lambda(4x + 3y - 19)` = where `lambda` is equal to

  • `0, - 40/9`

  • 0

  • `40/9`

  • `(- 40)/9`

Exercise 5.6 | Q 2 | Page 215

Choose the correct alternative:

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

  • `4/3`

  • `4/sqrt(3)`

  • `2/sqrt(3)`

  • `3/2`

Exercise 5.6 | Q 3 | Page 215

Choose the correct alternative:

The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if

  • 15 < m < 65

  • 35 < m < 85

  • – 85 < m < – 35

  • – 35 < m < 15

Exercise 5.6 | Q 4 | Page 215

Choose the correct alternative:

The length of the diameter of the circle which touches the x -axis at the point (1, 0) and passes through the point (2, 3)

  • `6/5`

  • `5/3`

  • `10/3`

  • `3/5`

Exercise 5.6 | Q 5 | Page 215

Choose the correct alternative:

The radius of the circle 3x2 + by2 + 4bx – 6by + b2 = 0 is

  • 1

  • 3

  • `sqrt(10)`

  • `sqrt(11)`

Exercise 5.6 | Q 6 | Page 215

Choose the correct alternative:

The centre of the circle inscribed in a square formed by the lines `x^2 - 8x - 12` = 0 and `y^2 - 14y + 45` = 0 is

  • (4, 7)

  • (7, 4)

  • (9, 4)

  • (4, 9)

Exercise 5.6 | Q 7 | Page 216

Choose the correct alternative:

The equation of the normal to the circle x2 + y2 – 2x – 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is

  • x + 2y = 3

  • x + 2y + 3 = 0

  • 2x + 4y + 3 = 0

  • x – 2y + 3 = 0

Exercise 5.6 | Q 8 | Page 216

Choose the correct alternative:

If P(x, y) be any point on 16x2 + 25y2 = 400 with foci F(3, 0) then PF1 + PF2 is

  • 8

  • 6

  • 10

  • 12

Exercise 5.6 | Q 9 | Page 216

Choose the correct alternative:

The radius of the circle passing through the points (6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is

  • 10

  • `2sqrt(5)`

  • 6

  • 4

Exercise 5.6 | Q 10 | Page 216

Choose the correct alternative:

The area of quadrilateral formed with foci of the hyperbolas `x^2/"a"^2 - y^2/"b"^2` = 1 and `1x^2/"a"^2 - y^2/"b"^2` = – 1

  • 4(a2 + b2)

  • 2(a2 + b2)

  • 2(a2 + b2)

  • `1/2 ("a"^2 + b"^2)`

Exercise 5.6 | Q 11 | Page 216

Choose the correct alternative:

If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x – 3)2 + (y + 2)2 = r2, then the value of r2 is

  • 2

  • 3

  • 1

  • 4

Exercise 5.6 | Q 12 | Page 216

Choose the correct alternative:

If x + y = k is a normal to the parabola y2 = 12x, then the value of k is 14

  • 3

  • – 1

  • 1

  • 9

Exercise 5.6 | Q 13 | Page 216

Choose the correct alternative:

The ellipse E1 : `x^2/9 + y^2/4` = 1 is inscribed in a rectangle R whose sides are parallel to the co-ordinate axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse is

  • `sqrt(2)/2`

  • `sqrt(3)/2`

  • `1/2`

  • `3/4`

Exercise 5.6 | Q 14 | Page 216

Choose the correct alternative:

Tangents are drawn to, the, hyperbola `x^2/9 - y^2/4` = 1 parallel to the straight line 2x – y – 1. One of the points of contact of tangents on the hyperbola is

  • `(9/(2sqrt(2)), (-1)/sqrt(2))`

  • `((9)/(2sqrt(2)), 1/sqrt(2))`

  • `(9/(2sqrt(2)), 1/sqrt(2))`

  • `(3sqrt(3), -2sqrt(2))`

Exercise 5.6 | Q 15 | Page 216

Choose the correct alternative:

The equation of the circle passing through the foci of the ellipse `x^2/16 +  y^2/9` = 1 having centre at (0, 3) is

  • x2 + y2 – 6y – 7 = 0

  • x2 + y2 – 6y + 7 = 0

  • x2 + y2 – 6y – 5 = 0

  • x2 + y2 – 6y + 5 = 0

Exercise 5.6 | Q 16 | Page 217

Choose the correct alternative:

Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centered at (0, y) passing through the origin and touching the circle C externally, then the radius of T is equal to

  • `sqrt(3)/sqrt(2)`

  • `sqrt(3)/2`

  • `1/2`

  • `1/4`

Exercise 5.6 | Q 17 | Page 217

Choose the correct alternative:

Consider an ellipse whose centre is of the origin and its major axis is a long x-axis. If its eccentricity is `3/5` and the distance between its foci is 6, then the area of the quadrilateral’ inscribed in the ellipse with diagonals as major and minor axis, of the ellipse is

  • 8

  • 32

  • 80

  • 40

Exercise 5.6 | Q 18 | Page 217

Choose the correct alternative:

Area of the greatest rectangle inscribed in the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 is

  • 2ab

  • ab

  • `sqrt("ab")`

  • `"a"/"b"`

Exercise 5.6 | Q 19 | Page 217

Choose the correct alternative:

An ellipse has OB as semi-minor axes, F and F’ its foci and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is

  • `1/sqrt(2)`

  • `1/2`

  • `1/4`

  • `1/sqrt(3)`

Exercise 5.6 | Q 20 | Page 217

Choose the correct alternative:

The eccentricity of the ellipse (x – 3)2 + (y – 4)2 = `y^2/9` is

  • `sqrt(3)/2`

  • `1/3`

  • `1/(3sqrt(2)`

  • `1/sqrt(3)`

Exercise 5.6 | Q 21 | Page 217

Choose the correct alternative:

If the two tangents drawn from a point P to the parabola y2 = 4r are at right angles then the locus of P is 

  • 2x + 1 = 0

  • x = – 1

  • 2x – 1 = 0

  • x = 1

Exercise 5.6 | Q 22 | Page 217

Choose the correct alternative:

The circle passing through (1, – 2) and touching the axis of x at (3, 0) passing through the point

  • (– 5, 2)

  • (2, – 5)

  • (5, – 2)

  • (– 2, 5)

Exercise 5.6 | Q 23 | Page 217

Choose the correct alternative:

The locus of a point whose distance from (– 2, 0) is `2/3` times its distance from the line x = `(-9)/2` is

  • a parabola

  • a hyperbola

  • an ellipse

  • a circle

Exercise 5.6 | Q 24 | Page 217

Choose the correct alternative:

The values of m for which the line y = `"m"x + 2sqrt(5)` touches the hyperbola 16x2 – 9y2 = 144 are the roots of x2 – (a + b)x – 4 = 0, then the value of (a + b) is

  • 2

  • 4

  • 0

  • – 2

Exercise 5.6 | Q 25 | Page 217

Choose the correct alternative:

If the coordinates at one end of a diameter of the circle x2 + y2 – 8x – 4y + c = 0 are (11, 2) the coordinates of the other end are

  • (– 5, 2)

  • (– 3, 2)

  • (5, – 2)

  • (– 2, 5)

Solutions for Chapter 5: Two Dimensional Analytical Geometry-II

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6
Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 5 - Two Dimensional Analytical Geometry-II - Shaalaa.com

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 5 - Two Dimensional Analytical Geometry-II

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 5 Two Dimensional Analytical Geometry-II are Two Dimensional Analytical Geometry-II, Circles, Conics, Conic Sections, Parametric Form of Conics, Tangents and Normals to Conics, Real Life Applications of Conics.

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