Question

Which of the following statement are true? Give reasons for your answers, either with a short proof or a counterexample.

i) Under a rotation of axes, a parabola can become a hyperbola.
ii) 1/sqrt(2), 1/sqrt(3)1/sqrt(5) form the direction cosines of a line.
iii) 2x + 3y = 7z = represents a line in three-dimensional space.

iv) x + y +z = 0 touches the cone x2 + y2 + z2 + 2(yz + zx + zy) = 0.

v) x = y = z - 1 does not intersect the cone x2 + y2 + z2 + 2(yz + zx+ xy) = 0 .

vi) Every cylinder has a circular base.

vii) Any curve in 2D that is symmetric about the origin is also symmetric with respect to both the axes.

viii) Given any two conics C1 and C2 , at least one element of
R x R  will belong to  C1 ∩ C2 .

ix) ax2 + by2 + cz2 = d  represents a sphere with radius sqrt(a2 + b2 +c2 - d), where a, b, c,d, are positive real numbers.

x) Every conicoid has a unique centre.

12 Feb 2018
Answer :
Word Count : 1089

Let's analyze each of the given statements about Analytical Geometry one by one:

i) Under a rotation of axes, a parabola can become a hyperbola.
   - False.
   - A parabola cannot become a hyperbola under a rotation of axes. A parabola is a conic section that can be defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). A hyperbola, on the other hand, is a conic section defined as the set of all points where the difference of distances to two fixed points (foci) is constant. These two curves have fundamentally different definitions and properties, and a rotation of axes cannot change one into the other.

ii) 1/sqrt(2), 1/sqrt(3), 1/sqrt(5) form the direction cosines of a line.
    - True.
    - The direction cosines of a line represent the cosines of the angles between the line and the coordinate axes. For a line with direction cosines (l, m, n), where l, m, and n are real numbers, the direction cosines satisfy the following:
      - l² + m² + n² = 1
    - In the given values, 1/sqrt(2), 1/sqrt(3), and 1/sqrt(5), the sum of their squares is indeed equal to 1:
      - (1/sqrt(2))² + (1/sqrt(3))² + (1/sqrt(5))² = 1/2 + 1/3 + 1/5 = (15 + 10 + 6) / 30 = 31/30
    - Since the sum of squares equals 1, these values can be direction cosines of a line.

iii) 2x + 3y = 7z represents a line in three-dimensional space.
     - True.
     - The equation 2x + 3y = 7z can be rewritten in vector form as:
       - r = <2, 3, -7> * t + <0, 0, 0>
     - This equation represents a line in three-dimensional space. The direction vector <2, 3, -7> determines the direction of the line, and any point on the line can be represented using the parameter

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Related Questions

If x/1 = y /1 = z/-1 represents one of the three mutually perpendicular generators of the cone 3xy + 8xz - 5yz =  0 , find the equations of the other two.
 

Check whether or not the conicoid represented by 5x2 + 4y2 - 4yz + 2xz + 2x - 4y - 8z + 2 = 0 is central or not. If it is, transform the equation by shifting the origin to the centre. Else, change any one coefficient to make the equation that of a central conicoid.

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Draw a rough sketch of the conic obtained in (i) above.

Which of the following statement are true? Give reasons for your answers, either with a short proof or a counterexample.

i) Under a rotation of axes, a parabola can become a hyperbola.
ii) 1/sqrt(2), 1/sqrt(3)1/sqrt(5) form the direction cosines of a line.
iii) 2x + 3y = 7z = represents a line in three-dimensional space.

iv) x + y +z = 0 touches the cone x2 + y2 + z2 + 2(yz + zx + zy) = 0.

v) x = y = z - 1 does not intersect the cone x2 + y2 + z2 + 2(yz + zx+ xy) = 0 .

vi) Every cylinder has a circular base.

vii) Any curve in 2D that is symmetric about the origin is also symmetric with respect to both the axes.

viii) Given any two conics C1 and C2 , at least one element of
R x R  will belong to  C1 ∩ C2 .

ix) ax2 + by2 + cz2 = d  represents a sphere with radius sqrt(a2 + b2 +c2 - d), where a, b, c,d, are positive real numbers.

x) Every conicoid has a unique centre.

Prove that the product of the distance from any point on a hyperbola to its asymptotes is a constant.

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