UET Lahore ECAT Mathematics: Analytic Geometry MCQs

Practice Analytic Geometry MCQs for UET Lahore ECAT Mathematics — topic-wise sets with solved answers.

UET Lahore ECAT Mathematics: Analytic Geometry MCQs — sample questions

  1. Question 1

    Q1. The distance between the points (2, 3) and (4, 5) is

    • A) √10
    • B) √8
    • C) √2
    • D) √20

    Answer: √2

    Explanation: Using distance formula: √((x2 - x1)² + (y2 - y1)²) = √((4 - 2)² + (5 - 3)²) = √(4 + 4) = √8 = 2√2

  2. Question 2

    Q2. The midpoint of the line segment joining (1, 2) and (3, 4) is

    • A) (2, 3)
    • B) (1, 1)
    • C) (4, 6)
    • D) (0, 0)

    Answer: (2, 3)

    Explanation: Using midpoint formula: ((x1 + x2) / 2, (y1 + y2) / 2) = ((1 + 3) / 2, (2 + 4) / 2) = (2, 3)

  3. Question 3

    Q3. The slope of the line passing through (2, 3) and (4, 5) is

    • A) 1
    • B) -1
    • C) 2
    • D) -2

    Answer: 1

    Explanation: Using slope formula: (y2 - y1) / (x2 - x1) = (5 - 3) / (4 - 2) = 2 / 2 = 1

  4. Question 4

    Q4. The equation of the circle with center (0, 0) and radius 4 is

    • A) x² + y² = 16
    • B) x² + y² = 4
    • C) x² - y² = 16
    • D) x² - y² = 4

    Answer: x² + y² = 16

    Explanation: Using circle equation: (x - h)² + (y - k)² = r², where (h, k) = (0, 0) and r = 4, so x² + y² = 16

  5. Question 5

    Q5. The equation of the line passing through (1, 2) with slope 3 is

    • A) y - 2 = 3(x - 1)
    • B) y + 2 = 3(x + 1)
    • C) y - 2 = -3(x - 1)
    • D) y + 2 = -3(x + 1)

    Answer: y - 2 = 3(x - 1)

    Explanation: Using point-slope form: y - y1 = m(x - x1), where m = 3, (x1, y1) = (1, 2), so y - 2 = 3(x - 1)

  6. Question 6

    Q6. The coordinates of the point dividing the line segment joining (1, 2) and (3, 4) in the ratio 2:1 are

    • A) (7 / 3, 10 / 3)
    • B) (5 / 3, 8 / 3)
    • C) (2, 3)
    • D) (4, 5)

    Answer: (7 / 3, 10 / 3)

    Explanation: Using section formula: ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)) = ((2*3 + 1*1) / (2 + 1), (2*4 + 1*2) / (2 + 1)) = (7 / 3, 10 / 3)

  7. Question 7

    Q7. The slope of the line 2x + 3y = 5 is

    • A) -2 / 3
    • B) 2 / 3
    • C) -3 / 2
    • D) 3 / 2

    Answer: -2 / 3

    Explanation: Converting to slope-intercept form: y = mx + b, where m = -A / B = -2 / 3

  8. Question 8

    Q8. The equation of the line parallel to y = 2x + 1 is

    • A) y = 2x + 3
    • B) y = -2x + 3
    • C) y = 1 / 2x + 3
    • D) y = -1 / 2x + 3

    Answer: y = 2x + 3

    Explanation: Parallel lines have the same slope, so the slope is 2, and the equation is of the form y = 2x + c

  9. Question 9

    Q9. The distance of the point (1, 2) from the line 3x + 4y = 5 is

    • A) |3 + 8 - 5| / √(3² + 4²)
    • B) |3 - 8 + 5| / √(3² + 4²)
    • C) |3 + 8 + 5| / √(3² + 4²)
    • D) |3 - 8 - 5| / √(3² + 4²)

    Answer: |3 + 8 - 5| / √(3² + 4²)

    Explanation: Using distance formula: |Ax1 + By1 - C| / √(A² + B²) = |3*1 + 4*2 - 5| / √(3² + 4²)

  10. Question 10

    Q10. The center and radius of the circle x² + y² - 4x - 6y = 12 are

    • A) (2, 3), √25
    • B) (-2, -3), √25
    • C) (2, 3), √12
    • D) (-2, -3), √12

    Answer: (2, 3), √25

    Explanation: Completing the square: (x - 2)² + (y - 3)² = 12 + 4 + 9 = 25, so center is (2, 3) and radius is √25 = 5

  11. Question 11

    Q11. The equation of the parabola with vertex (0, 0) and focus (0, 2) is

    • A) x² = 8y
    • B) x² = -8y
    • C) y² = 8x
    • D) y² = -8x

    Answer: x² = 8y

    Explanation: Using parabola equation: x² = 4ay, where a = 2, so x² = 8y

  12. Question 12

    Q12. The eccentricity of the ellipse 9x² + 4y² = 36 is

    • A) √5 / 3
    • B) √5 / 2
    • C) √3 / 2
    • D) √3 / 3

    Answer: √5 / 2

    Explanation: Converting to standard form: x² / 4 + y² / 9 = 1, where a² = 9, b² = 4, so e = √(1 - b² / a²) = √(1 - 4 / 9) = √5 / 3

  13. Question 13

    Q13. The equation of the ellipse with foci (0, ±3) and major axis 10 is

    • A) x² / 25 + y² / 16 = 1
    • B) x² / 16 + y² / 25 = 1
    • C) x² / 9 + y² / 25 = 1
    • D) x² / 25 + y² / 9 = 1

    Answer: x² / 16 + y² / 25 = 1

    Explanation: Using ellipse equation: c² = a² - b², where c = 3, 2a = 10, so a = 5, b² = 25 - 9 = 16, and the equation is x² / 16 + y² / 25 = 1

  14. Question 14

    Q14. The equation of the hyperbola with foci (0, ±5) and transverse axis 8 is

    • A) y² / 16 - x² / 9 = 1
    • B) y² / 9 - x² / 16 = 1
    • C) x² / 16 - y² / 9 = 1
    • D) x² / 9 - y² / 16 = 1

    Answer: y² / 16 - x² / 9 = 1

    Explanation: Using hyperbola equation: c² = a² + b², where c = 5, 2a = 8, so a = 4, b² = 25 - 16 = 9, and the equation is y² / 16 - x² / 9 = 1

  15. Question 15

    Q15. The angle between the lines 2x + y = 1 and x - 2y = 3 is

    • A) 90°
    • B) 60°
    • C) 45°
    • D) 30°

    Answer: 90°

    Explanation: Using angle formula: tanθ = |(m1 - m2) / (1 + m1*m2)|, where m1 = -2, m2 = 1 / 2, so tanθ is undefined, hence θ = 90°

  16. Question 16

    Q16. The equation of the tangent to the circle x² + y² = 4 at (2, 0) is

    • A) x = 2
    • B) y = 0
    • C) x + y = 2
    • D) x - y = 2

    Answer: x = 2

    Explanation: The tangent is perpendicular to the radius, so the slope of the tangent is undefined, hence the equation is x = 2

  17. Question 17

    Q17. The length of the latus rectum of the parabola y² = 8x is

    • A) 4
    • B) 8
    • C) 2
    • D) 1

    Answer: 8

    Explanation: Using parabola equation: y² = 4ax, where 4a = 8, so a = 2, and the length of the latus rectum is 4a = 8

  18. Question 18

    Q18. The equation of the director circle of the ellipse x² / 4 + y² / 9 = 1 is

    • A) x² + y² = 13
    • B) x² + y² = 5
    • C) x² + y² = 7
    • D) x² + y² = 11

    Answer: x² + y² = 13

    Explanation: Using director circle equation: x² + y² = a² + b², where a² = 9, b² = 4, so x² + y² = 13

  19. Question 19

    Q19. The asymptotes of the hyperbola x² / 4 - y² / 9 = 1 are

    • A) y = ±3 / 2x
    • B) y = ±2 / 3x
    • C) y = ±3x / 2
    • D) y = ±2x / 3

    Answer: y = ±3 / 2x

    Explanation: Using asymptotes equation: y = ±(b / a)x, where a = 2, b = 3, so y = ±(3 / 2)x

  20. Question 20

    Q20. The equation of the chord of contact of the tangents drawn from (2, 3) to the circle x² + y² = 4 is

    • A) 2x + 3y = 4
    • B) 2x - 3y = 4
    • C) x + y = 2
    • D) x - y = 2

    Answer: 2x + 3y = 4

    Explanation: Using chord of contact equation: T = 0, where T = xx1 + yy1 - r², so 2x + 3y = 4

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