20 more
Here are 20 more multiple choice questions (MCQs) on conic sections with explained answers:
1. The equation of the parabola with vertex at origin and directrix $$ y + 3 = 0 $$ is:
a) $$ y^2 = 12x $$
b) $$ (y - 3)^2 = 4ax $$
c) $$ (y + 3)^2 = 4ax $$
d) $$ y^2 = -12x $$
Answer: d) $$ y^2 = -12x $$
Explanation: The directrix is 3 units below the x-axis, so the parabola opens left with focal length 3.
2. The equation of parabola with focus at (-3,0) and directrix $$ x + 3 = 0 $$ is:
a) $$ y^2 = -12(x+3) $$
b) $$ y^2 = 12(x+3) $$
c) $$ x^2 = 12(y-3) $$
d) $$ y^2 = -12(x-3) $$
Answer: a) $$ y^2 = -12(x+3) $$
Explanation: Vertex midway between focus and directrix at (-3,0).
3. The length of the latus rectum of parabola $$ x^2 - 4x + 8y + 12 = 0 $$ is:
a) 2
b) 4
c) 6
d) 8
Answer: b) 4
Explanation: Standard form gives focal length 1, so latus rectum $$4a = 4$$.
4. The major axis of the ellipse $$ \frac{x^2}{25} + \frac{y^2}{9} = 1 $$ is along:
a) x-axis
b) y-axis
c) at 45 degrees
d) none
Answer: a) x-axis
Explanation: Larger denominator 25 is under $$x^2$$.
5. The eccentricity of ellipse $$ \frac{x^2}{25} + \frac{y^2}{16} = 1 $$ is:
a) 0.6
b) 0.8
c) 1.2
d) 1
Answer: a) 0.6
Explanation: $$e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{16}{25}} = 0.6$$.
6. The equation of a circle with center (2, -3) and radius 4 is:
a) $$ (x-2)^2 + (y+3)^2 = 16 $$
b) $$ (x+2)^2 + (y-3)^2 = 16 $$
c) $$ (x-2)^2 + (y-3)^2 = 16 $$
d) $$ (x+2)^2 + (y+3)^2 = 16 $$
Answer: a) $$ (x-2)^2 + (y+3)^2 = 16 $$
Explanation: Radius squared is $$4^2=16$$.
7. The hyperbola $$ \frac{x^2}{9} - \frac{y^2}{16} = 1 $$ opens:
a) horizontally
b) vertically
c) along y = x
d) none
Answer: a) horizontally
Explanation: Positive term $$x^2$$ is first, so opens along x-axis.
8. For the hyperbola $$ \frac{x^2}{9} - \frac{y^2}{16} = 1 $$, eccentricity is:
a) 5/3
b) 3/5
c) 1
d) 0
Answer: a) 5/3
Explanation: $$ e = \sqrt{1+\frac{b^2}{a^2}} = \sqrt{1 + \frac{16}{9}} = \frac{5}{3} $$.
9. The coordinates of foci of ellipse $$ \frac{x^2}{25} + \frac{y^2}{9} = 1 $$ are:
a) (±4,0)
b) (±5,0)
c) (±3,0)
d) (±0,±4)
Answer: a) (±4,0)
Explanation: $$c^2 = a^2 - b^2 = 25 -9 =16 \Rightarrow c=4$$.
10. The focus of parabola $$ y^2 = 12x $$ is at:
a) (3, 0)
b) (0, 3)
c) (6, 0)
d) (0, 6)
Answer: a) (3, 0)
Explanation: Focus at (a,0) with $$4a=12 \Rightarrow a=3$$.
11. The equation of the directrix of parabola $$ y^2 = 4ax $$ with a = 3 is:
a) $$ x = -3 $$
b) $$ x = 3 $$
c) $$ y = 3 $$
d) $$ y = -3 $$
Answer: a) $$ x = -3 $$
Explanation: Directrix for $$ y^2=4ax $$ is $$ x = -a $$.
12. The asymptotes of hyperbola $$ \frac{x^2}{9} - \frac{y^2}{16} = 1 $$ are:
a) $$ y = \pm \frac{4}{3}x $$
b) $$ y = \pm \frac{3}{4}x $$
c) $$ y = \pm 3x $$
d) $$ y = \pm 4x $$
Answer: a) $$ y = \pm \frac{4}{3}x $$
Explanation: Slopes of asymptotes are $$ \pm \frac{b}{a} $$.
13. The eccentricity of circle is:
a) 0
b) 1
c) between 0 and 1
d) greater than 1
Answer: a) 0
Explanation: Circle's eccentricity is zero.
14. The principal axis of ellipse $$ \frac{x^2}{49} + \frac{y^2}{25} = 1 $$ is:
a) 7
b) 14
c) 25
d) 49
Answer: b) 14
Explanation: Principal axis length equals $$2a = 2 \times 7 = 14$$.
15. The equation of tangent to parabola $$ y^2 = 4ax $$ at point P (at $$ t $$) is:
a) $$ y = tx + at^3 $$
b) $$ y = tx + at $$
c) $$ y = tx - at $$
d) $$ y = t^2 x + at $$
Answer: b) $$ y = tx + at $$
Explanation: Tangent form using parameter t.
16. The standard form of equation of ellipse with minor axis as x-axis is:
a) $$ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 $$
b) $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
c) $$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$
d) $$ \frac{x^2}{b^2} - \frac{y^2}{a^2} = 1 $$
Answer: a) $$ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 $$
Explanation: When minor axis is x-axis, $$a > b$$ and major axis along y-axis.
17. Eccentricity of parabola is:
a) 0
b) 1
c) between 0 and 1
d) more than 1
Answer: b) 1
Explanation: Parabola eccentricity equals 1.
18. For the hyperbola $$ \frac{y^2}{16} - \frac{x^2}{9} = 1 $$, the center is at:
a) (0,0)
b) (3,4)
c) (-3,-4)
d) (1,1)
Answer: a) (0,0)
Explanation: Standard hyperbola centered at origin.
19. The length of minor axis of ellipse $$ \frac{x^2}{16} + \frac{y^2}{25} = 1 $$ is:
a) 8
b) 10
c) 16
d) 50
Answer: a) 8
Explanation: Minor axis $$= 2b = 2 \times 4 = 8$$.
20. The parametric coordinates of point on ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ are:
a) $$ (a \cos \theta, b \sin \theta) $$
b) $$ (b \cos \theta, a \sin \theta) $$
c) $$ (a \sin \theta, b \cos \theta) $$
d) $$ (a \cot \theta, b \tan \theta) $$
Answer: a) $$ (a \cos \theta, b \sin \theta) $$
Explanation: Parameterization of ellipse.
21 to 40
Here are MCQs 21 to 40 on conic sections, with explained answers. These questions cover circles, parabolas, ellipses, and hyperbolas, continuing from the earlier ones.
***
### MCQs 21-40 on Conic Sections
#### 21. The equation $$y^2 = 8x$$ has its vertex at:
- a) (2, 0)
- b) (0, 2)
- c) (0, 0)
- d) (8, 0)
**Answer:** c) (0, 0)
**Explanation:** The standard form $$y^2 = 4ax$$ has its vertex at (0,0)[5].
***
#### 22. Which conic can have equal axes?
- a) Circle
- b) Ellipse
- c) Hyperbola
- d) Parabola
**Answer:** a) Circle
**Explanation:** In a circle, both axes (diameters) are equal[5].
***
#### 23. The length of major axis for ellipse $$\frac{x^2}{49} + \frac{y^2}{25} = 1$$ is:
- a) 10
- b) 12
- c) 14
- d) 7
**Answer:** c) 14
**Explanation:** Major axis length = $$2a$$; here $$a = 7$$, so length = 14[5].
***
#### 24. The eccentricity of a circle is:
- a) 1
- b) 0
- c) Between 0 and 1
- d) Infinity
**Answer:** b) 0
**Explanation:** By definition, the circle has eccentricity zero[5].
***
#### 25. The equation of a parabola with its axis along y-axis and vertex at origin is:
- a) x² = 4ay
- b) y² = 4ax
- c) x² = 4by
- d) y² = 4bx
**Answer:** a) x² = 4ay
**Explanation:** This is the standard form for such a parabola[5].
***
#### 26. The directrix of y² = -4ax is:
- a) x = a
- b) x = -a
- c) x = 0
- d) x = -a/2
**Answer:** a) x = a
**Explanation:** For y² = -4ax, the directrix is at x = a[5].
***
#### 27. What is the relation among axes for the standard hyperbola?
- a) Transverse axis > conjugate axis
- b) Transverse axis = conjugate axis
- c) Transverse axis < conjugate axis
- d) None of these
**Answer:** a) Transverse axis > conjugate axis
**Explanation:** In the standard hyperbola, the transverse axis is greater[5].
***
#### 28. A circle with center at (h, k) and radius r has the equation:
- a) (x + h)² + (y + k)² = r²
- b) (x - h)² + (y - k)² = r²
- c) x² + y² = r²
- d) None of these
**Answer:** b) (x - h)² + (y - k)² = r²
**Explanation:** That is the general form[5].
***
#### 29. Where are the foci of $$x^2/16 - y^2/9 = 1$$?
- a) (±4, 0)
- b) (±5, 0)
- c) (0, ±5)
- d) (0, ±3)
**Answer:** a) (±4, 0)
**Explanation:** For such a hyperbola, foci are at (±ae, 0); a=4, b=3, e=√(1+(9/16))[5].
***
#### 30. If the equation is $$4x^2 + 4y^2 - 8x + 12y - 25 = 0$$, the radius of the circle is:
- a) 2
- b) 3
- c) 4
- d) 5
**Answer:** b) 3
**Explanation:** $$g = -1, f = 1.5, c = -25/4$$; radius = $$\sqrt{g^2 + f^2 - c}$$[5].
***
#### 31. If a hyperbola has a transverse axis of 6, then a = ?
- a) 3
- b) 6
- c) 2
- d) 1
**Answer:** a) 3
**Explanation:** Transverse axis = 2a; thus a = 3[5].
***
#### 32. What is the sum of focal distances of any point on a hyperbola?
- a) 2a
- b) Constant
- c) Product equals 2a
- d) Not constant
**Answer:** d) Not constant
**Explanation:** In a hyperbola, their difference is constant but not their sum[5].
***
#### 33. For an ellipse, the eccentricity e is:
- a) > 1
- b) < 1
- c) = 1
- d) = 0
**Answer:** b) < 1
**Explanation:** By definition, for ellipse e < 1[5].
***
#### 34. A parabola opens upwards if equation is:
- a) y² = 4ax, a < 0
- b) x² = 4ay, a > 0
- c) x² = 4ay, a < 0
- d) y² = 4ax, a > 0
**Answer:** b) x² = 4ay, a > 0
**Explanation:** 'a' positive makes it open upwards[5].
***
#### 35. The latus rectum of hyperbola $$x^2/a^2 - y^2/b^2 = 1$$ is:
- a) b²/a
- b) 2b²/a
- c) 2a²/b
- d) a²/b
**Answer:** b) 2b²/a
**Explanation:** Latus rectum = $$2b²/a$$ for hyperbola[5].
***
#### 36. Equation x² + y² + 6x - 8y -11 = 0, center of circle?
- a) (-3, 4)
- b) (3, -4)
- c) (3, 4)
- d) (-3, -4)
**Answer:** a) (-3, 4)
**Explanation:** Center is at (-g, -f), here g = 3, f = -4[5].
***
#### 37. Focal length of parabola y² = 16x?
- a) 16
- b) 8
- c) 4
- d) 2
**Answer:** b) 8
**Explanation:** For y² = 4ax, focal length is 'a', so 4a = 16 ⇒ a = 4[5].
***
#### 38. Eccentricity of parabola y² = 4ax?
- a) 0
- b) 1
- c) Infinity
- d) Between 0 and 1
**Answer:** b) 1
**Explanation:** Parabola has e = 1[5].
***
#### 39. For ellipse, sum of focal distances is:
- a) a
- b) 2a
- c) b
- d) 2b
**Answer:** b) 2a
**Explanation:** Constant sum is always 2a[5].
***
#### 40. The equation $$y^2 - x = 0$$ is a parabola with vertex at:
- a) (0, 0)
- b) (1, 0)
- c) (0, 1)
- d) (1, 1)
**Answer:** a) (0, 0)
**Explanation:** Standard form y² = 4ax, vertex at origin[5].
***
41 to 60
Here are MCQs 41–60 on conic sections, each with an explained answer. These cover a variety of conceptual and application-based problems with circles, ellipses, parabolas, and hyperbolas.
***
### MCQs 41–60 on Conic Sections
#### 41. An arc of a bridge is semi-elliptical with a base length of 9 m and a height of 3 m at the center. The best approximation for the height of the arch 2 m from the center is:
- a) 2 m
- b) 2.5 m
- c) 2.75 m
- d) 2.9 m
**Answer:** b) 2.5 m
**Explanation:** Substitute $$x = 2$$ into the ellipse equation $$\frac{x^2}{(9/2)^2} + \frac{y^2}{3^2} = 1$$ and solve for y[1].
***
#### 42. The standard equation for a parabola with focus at (0, a), directrix y = –a, is:
- a) $$x^2 = 4a y$$
- b) $$y^2 = 4a x$$
- c) $$x^2 + y^2 = a^2$$
- d) $$x^2 - y^2 = a^2$$
**Answer:** a) $$x^2 = 4a y$$
**Explanation:** This is the equation for a parabola opening upward with given focus and directrix[1].
***
#### 43. The product of eccentricities of two conics is unity; one of them can be a/an:
- a) Parabola
- b) Ellipse
- c) Hyperbola
- d) Circle
**Answer:** c) Hyperbola
**Explanation:** For a hyperbola ($$e > 1$$), if another conic has $$e = 1/e_{hyperbola}$$, product is 1[1].
***
#### 44. If a tangent to the ellipse $$x^2/a^2 + y^2/b^2 = 1$$ at point P meets the coordinate axes at A and B, the minimum area of triangle OAB is:
- a) ab/2
- b) ab
- c) a^2b^2/4
- d) a^2b^2/2
**Answer:** a) ab/2
**Explanation:** Minimum area occurs when the tangent is equally inclined to axes. Use geometry of the ellipse[1].
***
#### 45. The conjugate axis of the hyperbola $$x^2/a^2 - y^2/b^2 = 1$$ is:
- a) Through center, perpendicular to transverse
- b) Focus
- c) Directrix
- d) Vertex
**Answer:** a) Through center, perpendicular to transverse
**Explanation:** By definition, the conjugate axis is perpendicular to the transverse axis[1].
***
#### 46. The length of transverse axis of the hyperbola $$x^2/9 - y^2/16 = 1$$ is:
- a) 8
- b) 6
- c) 12
- d) 18
**Answer:** b) 6
**Explanation:** Transverse axis = $$2a = 2 \times 3 = 6$$ for this equation[1].
***
#### 47. The eccentricity of the hyperbola $$x^2/a^2 - y^2/b^2 = 1$$ is:
- a) $$\sqrt{1 + b^2/a^2}$$
- b) $$\sqrt{a^2 + b^2}$$
- c) $$b/a$$
- d) None
**Answer:** a) $$\sqrt{1 + b^2/a^2}$$
**Explanation:** Eccentricity of a hyperbola is $$e = \sqrt{1 + \frac{b^2}{a^2}}$$[1].
***
#### 48. The locus of midpoints of chords of a hyperbola $$x^2/a^2 - y^2/b^2 = 1$$ parallel to the x-axis is:
- a) Another hyperbola
- b) Ellipse
- c) Parabola
- d) Straight line
**Answer:** d) Straight line
**Explanation:** Midpoints of parallel chords in conics are collinear[1].
***
#### 49. The area of the triangle formed by the lines $$x = 0, y = 0,$$ and $$x/a + y/b = 1$$ is:
- a) ab/2
- b) ab
- c) a^2b^2/2
- d) a^2b^2
**Answer:** a) ab/2
**Explanation:** Triangle vertices: (0, 0), (a, 0), (0, b). Area = $$\frac{1}{2} ab$$[1].
***
#### 50. The locus of center of circles passing through two fixed points is:
- a) Line
- b) Hyperbola
- c) Perpendicular bisector of their join
- d) Parabola
**Answer:** c) Perpendicular bisector of their join
**Explanation:** Centers must be equidistant from both points[1].
***
#### 51. The locus of intersection points of the pair of lines $$lx + my = n$$ as (l, m) varies is:
- a) A hyperbola of eccentricity 2
- b) Ellipse
- c) Parabola
- d) Circle
**Answer:** a) A hyperbola of eccentricity 2
**Explanation:** By calculation, this locus is a rectangular hyperbola[1].
***
#### 52. Equation $$x^2 + 4y^2 = 16$$ is:
- a) Circle
- b) Parabola
- c) Ellipse
- d) Hyperbola
**Answer:** c) Ellipse
**Explanation:** Two different squared terms, same sign; ellipse[6].
***
#### 53. A parabola with equation $$y^2 = 12x$$ has its latus rectum length:
- a) 3
- b) 6
- c) 12
- d) 24
**Answer:** c) 12
**Explanation:** Latus rectum of $$y^2 = 4ax$$ is $$4a = 12$$[6].
***
#### 54. The equation $$16x^2 + 25y^2 = 400$$ represents:
- a) Circle
- b) Parabola
- c) Ellipse
- d) Hyperbola
**Answer:** c) Ellipse
**Explanation:** Coefficients unequal, positive; ellipse centered at (0,0)[6].
***
#### 55. The minimum value of $$x^2 + y^2$$ subject to $$2x + 3y = 6$$ is:
- a) 10
- b) 4
- c) 6
- d) 9
**Answer:** a) 10
**Explanation:** The closest point from (0,0) on the line, found using method of least squares[6].
***
#### 56. Two conics intersect if their discriminant:
- a) $$\neq 0$$
- b) = 0
- c) Is positive
- d) None
**Answer:** a) $$\neq 0$$
**Explanation:** Non-zero discriminant means non-degenerate intersection[1].
***
#### 57. The common tangent to parabola $$y^2 = 4ax$$ and hyperbola $$x^2 - y^2 = a^2$$ is:
- a) $$y = x$$
- b) $$y = -x$$
- c) $$y = 0$$
- d) $$x = 0$$
**Answer:** a) $$y = x$$
**Explanation:** By comparison and substitution[1].
***
#### 58. The shortest distance between curves $$x^2 + y^2 = 1$$ and $$x^2 + y^2 = 4$$ is:
- a) 3
- b) 2
- c) 1
- d) 0
**Answer:** b) 1
**Explanation:** Difference of radii of concentric circles is $$2 - 1 = 1$$[1].
***
#### 59. The condition for a line with slope m to be normal to the parabola $$y^2 = 4ax$$ and tangent to the rectangular hyperbola $$xy = c^2$$ is:
- a) $$m^2 = 4a/c^2$$
- b) $$m^3 = 2ac$$
- c) $$m = c/a$$
- d) $$m^2 = a/c^2$$
**Answer:** a) $$m^2 = 4a/c^2$$
**Explanation:** Satisfies both the norm and tangent conditions for those conics[1].
***
#### 60. The locus of midpoints of chords of the hyperbola $$x^2 - y^2 = a^2$$ parallel to the x-axis is:
- a) y = constant
- b) x = constant
- c) Hyperbola
- d) Parabola
**Answer:** a) y = constant
**Explanation:** Chords parallel to x-axis have fixed y-values as midpoints[1].
***
