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.
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 t.
iv) x + y + z = 0 touches the cone x² + y² + z² + 2(yz + zx + zy) = 0.
- True.
- The equation x + y + z = 0 represents
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