LUMS LCAT Mathematics Analytic Geometry — Set 2

LUMS LCAT Mathematics Analytic Geometry — Set 2

1. Question 1

   Q1. The eccentricity of the ellipse x² / 4 + y² / 9 = 1 is

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

   Answer: √5 / 3

   Explanation: For the ellipse x² / b² + y² / a² = 1, where a > b, eccentricity e = √(1 - b² / a²) = √(1 - 4 / 9) = √(5 / 9) = √5 / 3

2. Question 2

   Q2. The angle between the lines y = x and y = -x is

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

   Answer: 90°

   Explanation: The slopes are 1 and -1, so the product of slopes is -1, indicating perpendicular lines, hence the angle is 90°

3. Question 3

   Q3. The equation of the ellipse with major axis 6 and minor axis 4 is

   • A) x² / 9 + y² / 4 = 1
   • B) x² / 4 + y² / 9 = 1
   • C) x² / 9 + y² / 4 = 0
   • D) x² / 4 + y² / 9 = 0

   Answer: x² / 9 + y² / 4 = 1

   Explanation: The general equation of an ellipse is x² / b² + y² / a² = 1, where 2a = 6 and 2b = 4, so a = 3 and b = 2

4. Question 4

   Q4. The focus of the parabola y² = 8x is

   • A) (2, 0)
   • B) (-2, 0)
   • C) (0, 2)
   • D) (0, -2)

   Answer: (2, 0)

   Explanation: The parabola y² = 4ax has focus (a, 0), here 4a = 8, so a = 2, hence focus is (2, 0)

5. Question 5

   Q5. The equation of the line passing through (1, 1) and parallel to y = 2x is

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

   Answer: y = 2x - 1

   Explanation: The slope of the line is 2, using point-slope form y - y1 = m(x - x1), we get y - 1 = 2(x - 1), simplifying to y = 2x - 1

6. Question 6

   Q6. The radius of the circle x² + y² + 2x + 2y + 1 = 0 is

   • A) 1
   • B) 2
   • C) √2
   • D) 1 / √2

   Answer: 1

   Explanation: The equation can be rewritten as (x + 1)² + (y + 1)² = 1, so the radius is √1 = 1

7. Question 7

   Q7. The distance of the point (3, 4) from the line 3x + 4y = 5 is

   • A) 1
   • B) 2
   • C) 3 / 5
   • D) 4 / 5

   Answer: 4 / 5

   Explanation: Using the distance formula: |Ax1 + By1 - C| / √(A² + B²) = |3*3 + 4*4 - 5| / √(3² + 4²) = |9 + 16 - 5| / √(9 + 16) = 20 / 5 = 4

8. Question 8

   Q8. The equation of the hyperbola with transverse axis 4 and conjugate axis 6 is

   • A) x² / 4 - y² / 9 = 1
   • B) x² / 9 - y² / 4 = 1
   • C) y² / 4 - x² / 9 = 1
   • D) y² / 9 - x² / 4 = 1

   Answer: x² / 4 - y² / 9 = 1

   Explanation: The general equation of a hyperbola is x² / a² - y² / b² = 1, where 2a = 4 and 2b = 6, so a = 2 and b = 3

9. Question 9

   Q9. The equation of the circle with diameter joining (1, 2) and (3, 4) is

   • A) (x - 2)² + (y - 3)² = 2
   • B) (x - 2)² + (y - 3)² = 4
   • C) (x - 1)² + (y - 2)² = 2
   • D) (x - 3)² + (y - 4)² = 2

   Answer: (x - 2)² + (y - 3)² = 2

   Explanation: The center is the midpoint (2, 3) and radius is half the distance between the points, which is √2, so the equation is (x - 2)² + (y - 3)² = 2

10. Question 10

    Q10. The length of the latus rectum of the parabola x² = 4y is

    • A) 2
    • B) 4
    • C) 6
    • D) 8

    Answer: 4

    Explanation: The length of the latus rectum of the parabola x² = 4ay is 4a, here 4a = 4, so a = 1 and length = 4

11. Question 11

    Q11. The equation of the tangent to the ellipse x² / 4 + y² / 9 = 1 at (0, 3) is

    • A) y = 3
    • B) y = -3
    • C) x = 0
    • D) x = 3

    Answer: y = 3

    Explanation: The tangent at (0, 3) is a horizontal line, so its equation is y = 3

12. Question 12

    Q12. The equation of the line passing through (2, 3) with slope 2 is

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

    Answer: y = 2x - 1

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

13. Question 13

    Q13. The length of the latus rectum of the parabola y² = 4ax is

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

    Answer: 4a

    Explanation: For parabola y² = 4ax, the length of the latus rectum is 4a, derived from the definition of the parabola.

14. Question 14

    Q14. The eccentricity of the ellipse x²/a² + y²/b² = 1 is

    • A) √(1 - b²/a²)
    • B) √(1 + b²/a²)
    • C) √(1 - a²/b²)
    • D) √(a² + b²)

    Answer: √(1 - b²/a²)

    Explanation: Eccentricity e = √(1 - b²/a²) for an ellipse, where a is the semi-major axis and b is the semi-minor axis.

15. Question 15

    Q15. The distance of the point (1, 2) from the line 3x + 4y + 5 = 0 is

    • A) 10/5
    • B) 12/5
    • C) 3
    • D) 4

    Answer: 12/5

    Explanation: Using the distance formula: |Ax1 + By1 + C|/√(A² + B²), where A = 3, B = 4, C = 5, (x1, y1) = (1, 2).

16. Question 16

    Q16. The equation of the ellipse with foci at (±c, 0) and major axis 2a is

    • A) x²/a² + y²/b² = 1
    • B) x²/b² + y²/a² = 1
    • C) x²/a² - y²/b² = 1
    • D) x²/b² - y²/a² = 1

    Answer: x²/a² + y²/b² = 1

    Explanation: The standard form of the ellipse equation with major axis along x-axis is x²/a² + y²/b² = 1, where b² = a² - c².

17. Question 17

    Q17. The vertex of the parabola y = x² + 2x + 1 is

    • A) (-1, 0)
    • B) (1, 0)
    • C) (0, 1)
    • D) (-1, 1)

    Answer: (-1, 0)

    Explanation: Using the vertex form y = a(x - h)² + k, where (h, k) is the vertex, completing the square gives y = (x + 1)², so vertex is (-1, 0).

18. Question 18

    Q18. The equation of the hyperbola with foci at (±c, 0) is

    • A) x²/a² + y²/b² = 1
    • B) x²/a² - y²/b² = 1
    • C) y²/a² - x²/b² = 1
    • D) y²/a² + x²/b² = 1

    Answer: x²/a² - y²/b² = 1

    Explanation: The standard form of the hyperbola equation with transverse axis along x-axis is x²/a² - y²/b² = 1, where b² = c² - a².

19. Question 19

    Q19. The radius of the circle x² + y² + 2x + 4y + 4 = 0 is

    • A) 1
    • B) 2
    • C) 3
    • D) 1/2

    Answer: 1

    Explanation: Completing the square gives (x + 1)² + (y + 2)² = 1, so the radius is √1 = 1.

20. Question 20

    Q20. The length of the major axis of the ellipse x²/25 + y²/9 = 1 is

    • A) 10
    • B) 5
    • C) 6
    • D) 8

    Answer: 10

    Explanation: The major axis is 2a, where a² = 25, so a = 5 and the length is 2*5 = 10.