Analytical Geometry – Introduction MCQs With Answer

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)|

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