ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Conic Sections (Set 1)

Take the ECAT Mathematics — Conic Sections (Set 1): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30…

What this test covers

• 30 ECAT Mathematics

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

1. Q1. The centre and radius of the circle x² + y² − 6x + 8y = 0 are:

   • A. (−3, 4) and 5
   • B. (3, −4) and 5 ✓
   • C. (3, −4) and 25
   • D. (6, −8) and 10

   Answer: B. (3, −4) and 5

   Explanation: (x − 3)² + (y + 4)² = 9 + 16 = 25, so the centre is (3, −4) and the radius 5.

2. Q2. The circle with centre (−3, 2) and radius 4 has the equation:

   • A. x² + y² + 6x − 4y + 3 = 0
   • B. x² + y² − 6x + 4y − 3 = 0
   • C. x² + y² + 6x − 4y − 16 = 0
   • D. x² + y² + 6x − 4y − 3 = 0 ✓

   Answer: D. x² + y² + 6x − 4y − 3 = 0

   Explanation: (x + 3)² + (y − 2)² = 16 expands to x² + y² + 6x − 4y + 13 − 16 = 0, i.e. x² + y² + 6x − 4y − 3 = 0.

3. Q3. The radius of the circle x² + y² + 4x − 2y − 4 = 0 is:

   • A. 2
   • B. √5
   • C. 3 ✓
   • D. 9

   Answer: C. 3

   Explanation: g = 2, f = −1, c = −4: r = √(g² + f² − c) = √(4 + 1 + 4) = 3.

4. Q4. The circle having (1, 2) and (5, 6) as ends of a diameter has equation:

   • A. (x + 3)² + (y + 4)² = 8
   • B. (x − 1)² + (y − 2)² = 32
   • C. (x − 3)² + (y − 4)² = 32
   • D. (x − 3)² + (y − 4)² = 8 ✓

   Answer: D. (x − 3)² + (y − 4)² = 8

   Explanation: Centre = midpoint (3, 4); radius = half of √(16 + 16) = √8, so r² = 8.

5. Q5. The equation of the tangent to x² + y² = 25 at the point (3, 4) is:

   • A. 3x + 4y = 5
   • B. 3x − 4y = 25
   • C. 3x + 4y = 25 ✓
   • D. 4x + 3y = 25

   Answer: C. 3x + 4y = 25

   Explanation: The tangent to x² + y² = a² at (x₁, y₁) is xx₁ + yy₁ = a², giving 3x + 4y = 25.

The other 25 questions

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6. Q6. The length of the tangent from the point (5, 5) to the circle x² + y² = 9 is:

   • A. √41
   • B. 41
   • C. √59
   • D. √50

7. Q7. The point (2, 3) with respect to the circle x² + y² = 16 lies:

   • A. Outside the circle
   • B. At the centre
   • C. On the circle
   • D. Inside the circle

8. Q8. The focus of the parabola y² = 8x is:

   • A. (0, 2)
   • B. (8, 0)
   • C. (2, 0)
   • D. (−2, 0)

9. Q9. The directrix of the parabola y² = 12x is:

   • A. x = 3
   • B. x = −12
   • C. y = −3
   • D. x = −3

10. Q10. The length of the latus rectum of the parabola x² = 16y is:

    • A. 4
    • B. 32
    • C. 16
    • D. 8

11. Q11. The focus of the parabola (y − 2)² = 4(x − 1) is:

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

12. Q12. The parabola x² = −8y opens:

    • A. To the right
    • B. Downward
    • C. To the left
    • D. Upward

13. Q13. The eccentricity of the ellipse x²/25 + y²/9 = 1 is:

    • A. 3/4
    • B. 3/5
    • C. 5/4
    • D. 4/5

14. Q14. The foci of the ellipse x²/25 + y²/16 = 1 are:

    • A. (±5, 0)
    • B. (0, ±3)
    • C. (±3, 0)
    • D. (±√41, 0)

15. Q15. The length of the major axis of the ellipse 4x² + 9y² = 36 is:

    • A. 6
    • B. 3
    • C. 4
    • D. 9

16. Q16. The length of the latus rectum of the ellipse x²/16 + y²/9 = 1 is:

    • A. 3/2
    • B. 9/4
    • C. 9/2
    • D. 8/3

17. Q17. The eccentricity of the hyperbola x²/16 − y²/9 = 1 is:

    • A. 3/4
    • B. 4/5
    • C. 5/3
    • D. 5/4

18. Q18. The foci of the hyperbola x²/9 − y²/16 = 1 are:

    • A. (±3, 0)
    • B. (0, ±5)
    • C. (±√7, 0)
    • D. (±5, 0)

19. Q19. The asymptotes of the hyperbola x²/4 − y²/9 = 1 are:

    • A. y = ±(9/4)x
    • B. y = ±3x
    • C. y = ±(2/3)x
    • D. y = ±(3/2)x

20. Q20. A conic section whose eccentricity is exactly 1 is:

    • A. A parabola
    • B. A hyperbola
    • C. A circle
    • D. An ellipse

21. Q21. The eccentricity of a rectangular hyperbola x² − y² = a² is:

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

22. Q22. The equation 4x² − 9y² = 36 represents:

    • A. A parabola
    • B. A hyperbola
    • C. A circle
    • D. An ellipse

23. Q23. The equation x² + 4y² − 2x = 3 represents:

    • A. An ellipse
    • B. A hyperbola
    • C. A circle
    • D. A parabola

24. Q24. The equation x² + y² + 2gx + 2fy + c = 0 represents a real circle (of positive radius) only if:

    • A. g² + f² − c > 0
    • B. c > 0
    • C. g = f
    • D. g² + f² + c > 0

25. Q25. The parabola with focus (0, 3) and directrix y = −3 is:

    • A. x² = −12y
    • B. x² = 12y
    • C. y² = 12x
    • D. x² = 6y

26. Q26. The ellipse with foci (±2, 0) and vertices (±4, 0) has equation:

    • A. x²/16 + y²/12 = 1
    • B. x²/4 + y²/16 = 1
    • C. x²/16 + y²/20 = 1
    • D. x²/16 + y²/4 = 1

27. Q27. For what values of c is the line y = x + c tangent to the circle x² + y² = 8?

    • A. c = ±8
    • B. c = ±2
    • C. c = ±4
    • D. c = ±2√2

28. Q28. The line y = 2x + c touches the parabola y² = 8x when c equals:

    • A. 2
    • B. 1
    • C. 4
    • D. 1/2

29. Q29. The circles x² + y² = 4 and x² + y² − 6x + 5 = 0:

    • A. Touch externally
    • B. Touch internally
    • C. Do not meet
    • D. Intersect at two points

30. Q30. The curve x = 4 cos θ, y = 2 sin θ is:

    • A. The ellipse x²/16 + y²/4 = 1
    • B. The ellipse x²/4 + y²/16 = 1
    • C. The circle x² + y² = 16
    • D. The hyperbola x²/16 − y²/4 = 1