Analytical Geometry – Introduction MCQs With Answer provides B. Pharm students a focused review of coordinate geometry concepts essential for pharmaceutical calculations. This introduction covers distance and midpoint formulas, slope and equation of lines, circle and conic-section basics, and applied problems like geometric models for tablet design and diffusion gradients. These concise, keyword-rich MCQs help pharmacy undergraduates strengthen problem-solving skills used in dosage form geometry, spatial modeling, and quality control. Each question emphasizes clarity, stepwise reasoning, and real-world relevance to pharmacy studies. ‘Now let’s test your knowledge with 50 MCQs on this topic.’
Q1. What is the distance between points (2, 3) and (7, 11)?
- 9
- 10
- √53
- √85
Correct Answer: √53
Q2. The midpoint of the segment joining (−4, 5) and (6, −3) is:
- (1, 1)
- (−1, 2)
- (1, −1)
- (0, 0)
Correct Answer: (1, 1)
Q3. The slope of the line passing through (1, 2) and (4, 8) is:
- 2
- 3
- 6
- 1/3
Correct Answer: 2
Q4. Which is the slope-intercept form of a line?
- Ax + By + C = 0
- y = mx + c
- (y − y1) = m(x − x1)
- x/a + y/b = 1
Correct Answer: y = mx + c
Q5. The equation of a line with slope 3 passing through (2, −1) is:
- y = 3x + 5
- y + 1 = 3(x − 2)
- 3x − y − 7 = 0
- y = x/3 − 1
Correct Answer: y + 1 = 3(x − 2)
Q6. Two lines are perpendicular if the product of their slopes is:
- 1
- −1
- 0
- Undefined
Correct Answer: −1
Q7. Which condition indicates two lines are parallel?
- Their slopes are negative reciprocals
- Their slopes are equal
- Their y-intercepts are equal
- They intersect at origin
Correct Answer: Their slopes are equal
Q8. Equation of a circle with center (3, −2) and radius 5 is:
- (x − 3)^2 + (y + 2)^2 = 25
- (x + 3)^2 + (y − 2)^2 = 25
- (x − 3)^2 − (y + 2)^2 = 25
- x^2 + y^2 − 6x + 4y = 0
Correct Answer: (x − 3)^2 + (y + 2)^2 = 25
Q9. For circle x^2 + y^2 − 4x + 6y − 11 = 0, the center is:
- (2, −3)
- (−2, 3)
- (2, 3)
- (−2, −3)
Correct Answer: (2, −3)
Q10. Radius of the circle x^2 + y^2 + 8x − 6y + 9 = 0 is:
- 5
- 6
- √70
- √10
Correct Answer: 5
Q11. The general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 represents a parabola when:
- B^2 − 4AC < 0
- B^2 − 4AC = 0
- B^2 − 4AC > 0
- A + C = 0
Correct Answer: B^2 − 4AC = 0
Q12. Standard equation of a parabola with vertex at origin and focus at (a, 0) is:
- y^2 = 4ax
- x^2 = 4ay
- xy = a^2
- x^2 + y^2 = a^2
Correct Answer: y^2 = 4ax
Q13. Length of the latus rectum of parabola y^2 = 12x is:
- 12
- 6
- 3
- 4
Correct Answer: 12
Q14. The eccentricity of an ellipse lies between:
- 0 and 1
- 1 and ∞
- −1 and 0
- Equal to 0 only
Correct Answer: 0 and 1
Q15. Standard equation of an ellipse with major axis along x-axis is:
- x^2/a^2 + y^2/b^2 = 1 with a > b
- x^2/a^2 − y^2/b^2 = 1
- y^2/a^2 + x^2/b^2 = 1 with a > b
- x^2 + y^2 = r^2
Correct Answer: x^2/a^2 + y^2/b^2 = 1 with a > b
Q16. For hyperbola x^2/a^2 − y^2/b^2 = 1, its asymptotes are:
- y = ±(b/a)x
- y = ±(a/b)x
- y = ±ax + b
- y = ±bx + a
Correct Answer: y = ±(b/a)x
Q17. The distance from point (x0, y0) to line ax + by + c = 0 is given by:
- |ax0 + by0 + c|/(a + b)
- |ax0 + by0 + c|/√(a^2 + b^2)
- (ax0 + by0 + c)/√(a^2 + b^2)
- √(ax0^2 + by0^2 + c^2)
Correct Answer: |ax0 + by0 + c|/√(a^2 + b^2)
Q18. Three points (1,2), (3,6), (5,10) are:
- Collinear
- Vertices of a right triangle
- Vertices of an isosceles triangle
- Non-collinear
Correct Answer: Collinear
Q19. Area of triangle with vertices (0,0), (4,0), (0,3) is:
- 6
- 12
- 2
- 7
Correct Answer: 6
Q20. The equation of the perpendicular bisector of segment joining (2, 3) and (6, 7) is:
- x + y − 5 = 0
- x − y = 0
- x + y − 9 = 0
- x − y − 1 = 0
Correct Answer: x + y − 5 = 0
Q21. Coordinates of centroid of triangle with vertices (0,0), (3,0), (0,6) are:
- (1, 2)
- (1, 3)
- (3, 2)
- (0, 0)
Correct Answer: (1, 2)
Q22. If slope of line AB is 2 and slope of BC is −1/2, then AB is:
- Parallel to BC
- Perpendicular to BC
- Coincident with BC
- Neither parallel nor perpendicular
Correct Answer: Perpendicular to BC
Q23. The locus of points equidistant from point (0, c) and line y = −c (c > 0) is a:
- Circle
- Parabola
- Line
- Hyperbola
Correct Answer: Parabola
Q24. In coordinate geometry, the direction ratio of a line parallel to vector (2, −3) is:
- (3, 2)
- (2, −3)
- (−2, 3)
- (−3, −2)
Correct Answer: (2, −3)
Q25. The equation 3x + 4y = 12 has x-intercept and y-intercept respectively:
- (4, 0) and (0, 3)
- (3, 0) and (0, 4)
- (12, 0) and (0, 12)
- (−4, 0) and (0, −3)
Correct Answer: (4, 0) and (0, 3)
Q26. If the equation of line is y − 2 = m(x − 1) and it passes through (4, 8), m equals:
- 2
- 3
- 5/3
- 6/3
Correct Answer: 2
Q27. Condition for three points (x1,y1), (x2,y2), (x3,y3) to be collinear using determinant is:
- Determinant equals 1
- Determinant equals 0
- Determinant equals −1
- Determinant equals 2
Correct Answer: Determinant equals 0
Q28. The slope of tangent to circle x^2 + y^2 = r^2 at (x1, y1) is:
- −x1/y1
- y1/x1
- −y1/x1
- x1/y1
Correct Answer: −x1/y1
Q29. For parabola y^2 = 4ax, the parametric coordinates of a point are:
- (at^2, 2at)
- (at, at^2)
- (2at, at^2)
- (at^2, at)
Correct Answer: (at^2, 2at)
Q30. The focus of parabola y^2 = 8x is located at:
- (2, 0)
- (4, 0)
- (1, 0)
- (0, 2)
Correct Answer: (2, 0)
Q31. The eccentricity of hyperbola x^2/a^2 − y^2/b^2 = 1 is:
- c/a where c^2 = a^2 + b^2
- c/a where c^2 = a^2 − b^2
- b/a where b^2 = a^2 + c^2
- a/c where c^2 = b^2 − a^2
Correct Answer: c/a where c^2 = a^2 + b^2
Q32. The general condition to remove the xy-term by rotation is to choose angle θ such that:
- tan2θ = B/(A − C)
- tan2θ = 2B/(A − C)
- tanθ = B/(A + C)
- tanθ = 2B/(A + C)
Correct Answer: tan2θ = 2B/(A − C)
Q33. For ellipse x^2/9 + y^2/4 = 1, semi-major and semi-minor axes are:
- a = 3, b = 2
- a = 2, b = 3
- a = 9, b = 4
- a = 4, b = 9
Correct Answer: a = 3, b = 2
Q34. Which of the following represents a circle in general second-degree form?
- x^2 + y^2 + 4x − 6y + 9 = 0
- x^2 − y^2 + 2x + 3 = 0
- xy + x + y + 1 = 0
- x^2 + 2xy + y^2 + 1 = 0
Correct Answer: x^2 + y^2 + 4x − 6y + 9 = 0
Q35. If a line has equation 2x − y + 3 = 0, its slope is:
- 2
- −2
- 1/2
- −1/2
Correct Answer: 2
Q36. Which formula gives the angle θ between two lines with slopes m1 and m2?
- tan θ = |(m1 − m2)/(1 + m1 m2)|
- tan θ = (m1 + m2)/(1 − m1 m2)
- tan θ = |(m1 + m2)/(1 − m1 m2)|
- tan θ = (m1 − m2)/(1 − m1 m2)
Correct Answer: tan θ = |(m1 − m2)/(1 + m1 m2)|
Q37. In pharmacy formulation, approximating a tablet as a cylinder, volume is given by:
- V = πr^2h
- V = 2πr^2h
- V = πr h
- V = 2πrh
Correct Answer: V = πr^2h
Q38. The chord joining points where line y = mx + c meets circle x^2 + y^2 = r^2 is bisected at:
- The line passes through origin
- The midpoint lies on the line perpendicular to chord through center
- The midpoint lies at center only
- Midpoint equals (m, c)
Correct Answer: The midpoint lies on the line perpendicular to chord through center
Q39. The equation x^2 + 2xy + y^2 = 0 represents:
- Pair of real coincident lines
- Pair of straight lines (x + y)^2 = 0
- Hyperbola
- Circle
Correct Answer: Pair of straight lines (x + y)^2 = 0
Q40. The slope of normal to curve at a point is the negative reciprocal of:
- Slope of tangent at that point
- Slope of radius only
- Slope of chord joining two points
- None of the above
Correct Answer: Slope of tangent at that point
Q41. Coordinates of the circumcenter of triangle with vertices (0,0), (a,0), (0,b) are:
- ((a/2), (b/2))
- ((a+b)/3, 0)
- ((a^2)/(2a), (b^2)/(2b))
- ((a/2), (b/2)) if triangle is right-angled at origin
Correct Answer: ((a/2), (b/2)) if triangle is right-angled at origin
Q42. If point P(x, y) divides the line joining (x1, y1) and (x2, y2) internally in ratio m:n, P is given by:
- ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
- ((mx1 + nx2)/(m+n), (my1 + ny2)/(m+n))
- ((x1 + x2)/2, (y1 + y2)/2)
- ((mx1 − nx2)/(m−n), (my1 − ny2)/(m−n))
Correct Answer: ((mx1 + nx2)/(m+n), (my1 + ny2)/(m+n))
Q43. Which quantity remains constant along any circle centered at origin?
- x + y
- x^2 + y^2
- x − y
- xy
Correct Answer: x^2 + y^2
Q44. The locus of midpoints of parallel chords of a circle passes through:
- The center of the circle
- A fixed diameter line parallel to chords
- A parabola
- No fixed line
Correct Answer: The center of the circle
Q45. For line 4x − 3y + 12 = 0, the perpendicular distance from (1, 2) to the line is:
- |4(1) − 3(2) + 12|/5 = |10|/5 = 2
- |4 − 6 + 12|/√(25) = 10/5 = 2
- |4 − 6 + 12|/5 = 10/5 = 2
- |4 − 6 + 12|/√(16 + 9) = 10/√25 = 2
Correct Answer: |4 − 6 + 12|/√(16 + 9) = 10/√25 = 2
Q46. Which of these is true for any parabola y^2 = 4ax?
- Directrix is x = a
- Focus is at (a, 0)
- Axis is y = 0
- Directrix is x = −a
Correct Answer: Directrix is x = −a
Q47. In analytic geometry, the term “locus” refers to:
- A fixed point
- A set of points satisfying a condition
- A single line only
- A circle only
Correct Answer: A set of points satisfying a condition
Q48. Which of the following is a property of the midpoint of a chord of a circle that passes through the center?
- It is equidistant from the circle’s circumference on both sides
- It lies at the center
- It bisects the chord and lies on a diameter
- It is always at origin
Correct Answer: It bisects the chord and lies on a diameter
Q49. For hyperbola xy = c^2 (rectangular hyperbola), asymptotes are:
- x = 0 and y = 0
- y = ±x
- x + y = 0 only
- None of the above
Correct Answer: x = 0 and y = 0
Q50. Which formula is useful for finding area of triangle with vertices (x1,y1), (x2,y2), (x3,y3)?
- Area = 1/2 |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|
- Area = (x1 + x2 + x3)(y1 + y2 + y3)/2
- Area = 1/2 (distance between x-coordinates)*(distance between y-coordinates)
- Area = |(x1y1 + x2y2 + x3y3)|
Correct Answer: Area = 1/2 |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|
