TRB BT Assistant - Pole, Polar and Conjugate Elements Online Test (MCQ Quiz)

1. What is the locus of points the product of whose distances from two fixed points is constant?

A) Ellipse B) Hyperbola C) Parabola D) Circle

2. What is the locus of points whose distance from a fixed point (focus) is proportional to its distance from a fixed line (directrix)?

A) Ellipse B) Hyperbola C) Parabola D) Circle

3. What is the locus of points equidistant from a fixed point and a fixed line?

A) Ellipse B) Hyperbola C) Parabola D) Circle

4. In the context of conic sections, what is the eccentricity 'e' for an ellipse?

A) e = 0 B) 0 < e < 1 C) e = 1 D) e > 1

5. In the context of conic sections, what is the eccentricity 'e' for a parabola?

A) e = 0 B) 0 < e < 1 C) e = 1 D) e > 1

6. In the context of conic sections, what is the eccentricity 'e' for a hyperbola?

A) e = 0 B) 0 < e < 1 C) e = 1 D) e > 1

7. What is the eccentricity 'e' for a circle?

A) e = 0 B) 0 < e < 1 C) e = 1 D) e > 1

8. What is the definition of the pole of a point with respect to a conic section?

A) The intersection of tangents drawn from the point to the conic. B) The line joining the points of intersection of any secant through the point with the conic. C) The locus of points from which tangents to the conic are perpendicular. D) The line such that the given point is the pole of this line with respect to the conic.

9. What is the definition of the polar of a point with respect to a conic section?

A) The intersection of tangents drawn from the point to the conic. B) The line joining the points of intersection of any secant through the point with the conic. C) The locus of points from which tangents to the conic are perpendicular. D) The line such that the given point is the pole of this line with respect to the conic.

10. If the polar of a point P with respect to a conic passes through a point Q, then the polar of Q passes through which point?

A) The center of the conic B) The point P C) The focus of the conic D) The vertex of the conic

11. When is the polar of a point P with respect to a conic a tangent to the conic?

A) When P is at infinity. B) When P is inside the conic. C) When P is outside the conic. D) When P lies on the conic.

12. What is the equation of the polar of a point (x1, y1) with respect to the circle x^2 + y^2 = a^2?

A) xx1 + yy1 = a^2 B) xx1 - yy1 = a^2 C) x + x1 = a^2 D) y + y1 = a^2

13. What is the equation of the polar of a point (x1, y1) with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1?

A) (xx1/a^2) + (yy1/b^2) = 1 B) (xx1/a^2) - (yy1/b^2) = 1 C) (x/a^2) + (y/b^2) = 1 D) (x1/a^2) + (y1/b^2) = 1

14. What is the equation of the polar of a point (x1, y1) with respect to the parabola y^2 = 4ax?

A) yy1 = 2a(x + x1) B) yy1 = 2a(x - x1) C) yy1 = 2a(x1 - x) D) y = 2a(x + x1)

15. What is the equation of the polar of a point (x1, y1) with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1?

A) (xx1/a^2) - (yy1/b^2) = 1 B) (xx1/a^2) + (yy1/b^2) = 1 C) (x/a^2) - (y/b^2) = 1 D) (x1/a^2) - (y1/b^2) = 1

16. What is the pole of the line Ax + By + C = 0 with respect to the circle x^2 + y^2 = a^2?

A) (-a^2A/C, -a^2B/C) B) (a^2A/C, a^2B/C) C) (-a^2C/A, -a^2C/B) D) (a^2C/A, a^2C/B)

17. What is the pole of the line Ax + By + C = 0 with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1?

A) (-a^2A/C, -b^2B/C) B) (a^2A/C, b^2B/C) C) (-a^2C/A, -b^2C/B) D) (a^2C/A, b^2C/B)

18. What is the pole of the line Ax + By + C = 0 with respect to the parabola y^2 = 4ax?

A) (aB^2/AC, -2aB/A) B) (-aB^2/AC, 2aB/A) C) (2aB/A, aB^2/AC) D) (-2aB/A, aB^2/AC)

19. What is the pole of the line Ax + By + C = 0 with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1?

A) (a^2A/C, -b^2B/C) B) (-a^2A/C, b^2B/C) C) (a^2C/A, -b^2C/B) D) (-a^2C/A, b^2C/B)

20. Two points P and Q are said to be conjugate with respect to a conic if the polar of P passes through which point?

A) The center of the conic B) The point Q C) The focus of the conic D) The vertex of the conic

21. If P and Q are conjugate points with respect to a conic, then which of the following is true?

A) The polar of Q passes through P. B) The polar of P is perpendicular to the polar of Q. C) The line PQ passes through the center of the conic. D) The line PQ is tangent to the conic.

22. What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the circle x^2 + y^2 = a^2?

A) x1x2 + y1y2 = a^2 B) x1x2 + y1y2 = 0 C) x1x2 - y1y2 = a^2 D) x1x2 - y1y2 = 0

23. What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1?

A) (x1x2/a^2) + (y1y2/b^2) = 1 B) (x1x2/a^2) + (y1y2/b^2) = 0 C) (x1x2/a^2) - (y1y2/b^2) = 1 D) (x1x2/a^2) - (y1y2/b^2) = 0

24. What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the parabola y^2 = 4ax?

A) y1y2 = 2a(x1 + x2) B) y1y2 = 2a(x1 - x2) C) y1y2 = 2a(x2 - x1) D) y1y2 = 4a(x1 + x2)

25. What is the condition for two points (x1, y1) and (x2, y2) to be conjugate with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1?

A) (x1x2/a^2) - (y1y2/b^2) = 1 B) (x1x2/a^2) - (y1y2/b^2) = 0 C) (x1x2/a^2) + (y1y2/b^2) = 1 D) (x1x2/a^2) + (y1y2/b^2) = 0

26. Two lines L1 and L2 are said to be conjugate with respect to a conic if each passes through the pole of the other. If L1 is y = mx + c, what is the condition for L2 to be conjugate to L1?

A) L2 passes through the center of the conic. B) L2 is parallel to L1. C) The pole of L1 lies on L2. D) The pole of L2 lies on L1.

27. What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the circle x^2 + y^2 = a^2?

A) m1m2 = -1 B) m1 + m2 = 0 C) c1c2 = a^2(1 + m1m2) D) c1c2 = -a^2(1 + m1m2)

28. What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the ellipse (x^2/a^2) + (y^2/b^2) = 1?

A) m1m2 = -b^2/a^2 B) m1m2 = -a^2/b^2 C) b^2 + m1m2a^2 = 0 D) a^2 + m1m2b^2 = 0

29. What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the parabola y^2 = 4ax?

A) m1m2 = -1 B) m1 + m2 = 0 C) am1m2 = c1 + c2 D) am1m2 = c1 - c2

30. What is the condition for the lines y = m1x + c1 and y = m2x + c2 to be conjugate with respect to the hyperbola (x^2/a^2) - (y^2/b^2) = 1?

A) m1m2 = b^2/a^2 B) m1m2 = -b^2/a^2 C) b^2 - m1m2a^2 = 0 D) a^2 - m1m2b^2 = 0

31. If the tangents at points P and Q on a conic intersect at T, then the polar of T is the line joining which points?

A) The center and the vertex B) The foci C) The points P and Q D) The intersection points of the conic with the line joining the center to T

32. The locus of the intersection of perpendicular tangents to a conic is called:

A) Director circle B) Directrix C) Polar D) Axis

33. What is the director circle of the ellipse (x^2/a^2) + (y^2/b^2) = 1?

A) x^2 + y^2 = a^2 + b^2 B) x^2 + y^2 = a^2 - b^2 C) x^2 + y^2 = a^2 D) x^2 + y^2 = b^2

34. What is the director circle of the hyperbola (x^2/a^2) - (y^2/b^2) = 1?

A) x^2 + y^2 = a^2 + b^2 B) x^2 + y^2 = a^2 - b^2 C) x^2 + y^2 = a^2 D) x^2 + y^2 = b^2

35. What is the director curve of the parabola y^2 = 4ax?

A) The directrix B) The axis of symmetry C) The tangent at the vertex D) A point at infinity

36. If the polars of two points P and Q with respect to a conic are perpendicular, are P and Q necessarily conjugate?

A) Yes B) No C) Only if the conic is a circle D) Only if the conic is a parabola

37. Consider the conic Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0. If the polar of (x1, y1) is L, then the pole of L is:

A) (x1, y1) B) The center of the conic C) The origin D) Infinity

38. What is the condition for the line lx + my + n = 0 to be a tangent to the ellipse (x^2/a^2) + (y^2/b^2) = 1?

A) a^2l^2 + b^2m^2 = n^2 B) a^2m^2 + b^2l^2 = n^2 C) a^2l^2 - b^2m^2 = n^2 D) a^2m^2 - b^2l^2 = n^2

39. What is the condition for the line lx + my + n = 0 to be a tangent to the parabola y^2 = 4ax?

A) a*l^2 = m*n B) a*m^2 = l*n C) a*l^2 = -m*n D) a*m^2 = -l*n

40. What is the condition for the line lx + my + n = 0 to be a tangent to the hyperbola (x^2/a^2) - (y^2/b^2) = 1?

A) a^2l^2 - b^2m^2 = n^2 B) a^2m^2 - b^2l^2 = n^2 C) a^2l^2 + b^2m^2 = n^2 D) a^2m^2 + b^2l^2 = n^2

41. The tangents from an external point to a conic touch the conic at points whose join is the:

A) Polar of the external point B) Polar of the center C) Directrix D) Axis

42. If the polars of P and Q with respect to a central conic intersect at O, then O is:

A) The center of the conic B) The pole of PQ C) The focus D) The vertex

43. What is the pole of the line Ax + By + C = 0 with respect to the general conic Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0?

A) The solution of the system of equations: Ax + Hy + G = -C(Ax+Hy+G)/(Ax1+Hy1+G) B) The solution of the system of equations: Ax + Hy + G = -C(lx+my+n)/(Al+Hm+G) C) The solution of the system of equations: Ax + Hy + G = -n(Al+Hm+G)/C D) The solution of the system of equations: Gx + Fy + C = 0

44. What is the polar of the origin (0,0) with respect to the general conic Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0?

A) 2Gx + 2Fy + C = 0 B) Gx + Fy + C = 0 C) Ax + Hy + G = 0 D) Hx + By + F = 0

45. Two chords of a conic passing through a point P are divided harmonically by P and the polar of P. This is known as the theorem of:

A) Apollonius B) Pascal C) Brianchon D) Harmonic conjugates

46. If P and Q are conjugate points with respect to a central conic, and the line PQ intersects the conic at A and B, then the tangents at A and B intersect on:

A) The line PQ B) The polar of P C) The polar of Q D) The line joining the center to P

47. The locus of the poles of a system of parallel chords of a conic is a line passing through:

A) The center of the conic B) The focus C) The vertex D) The intersection of the directrix and axis

48. If the polars of P and Q with respect to a conic are the same line L, then P and Q are:

A) Conjugate points B) The same point C) At infinity D) The foci

49. The polar of a point on a conic is:

A) The tangent at that point B) The normal at that point C) The chord of contact D) The directrix

50. The pole of the directrix of a conic section is located at:

A) The focus B) The center C) The vertex D) Infinity

Pole, Polar and Conjugate Elements - Online Test

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