ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Conic Sections (Set 2)

Take the ECAT Mathematics — Conic Sections (Set 2): 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 + 4x − 6y − 3 = 0 are:

   • A. (−2, 3) and 4 ✓
   • B. (−4, 6) and √3
   • C. (−2, 3) and 16
   • D. (2, −3) and 4

   Answer: A. (−2, 3) and 4

   Explanation: Centre (−g, −f) = (−2, 3); r = √(g² + f² − c) = √(4 + 9 + 3) = 4.

2. Q2. The circle with centre (2, −1) passing through (5, 3) is:

   • A. (x − 2)² + (y + 1)² = 5
   • B. (x − 5)² + (y − 3)² = 25
   • C. (x + 2)² + (y − 1)² = 25
   • D. (x − 2)² + (y + 1)² = 25 ✓

   Answer: D. (x − 2)² + (y + 1)² = 25

   Explanation: r² = (5 − 2)² + (3 + 1)² = 9 + 16 = 25, with centre (2, −1).

3. Q3. The circle centred at the origin that touches the line 3x + 4y = 10 is:

   • A. x² + y² = 2
   • B. x² + y² = 100
   • C. x² + y² = 10
   • D. x² + y² = 4 ✓

   Answer: D. x² + y² = 4

   Explanation: The radius equals the distance from O to the line: 10/√(9 + 16) = 2, so x² + y² = 4.

4. Q4. The tangent to x² + y² = 13 at the point (2, −3) is:

   • A. 2x − 3y = 13 ✓
   • B. 2x + 3y = 13
   • C. 2x − 3y = 0
   • D. 3x − 2y = 13

   Answer: A. 2x − 3y = 13

   Explanation: Tangent at (x₁, y₁) to x² + y² = a² is xx₁ + yy₁ = a²: 2x − 3y = 13.

5. Q5. The length of the tangent from (4, 3) to the circle x² + y² − 2x − 4 = 0 is:

   • A. √13 ✓
   • B. 13
   • C. 3
   • D. √21

   Answer: A. √13

   Explanation: Length = √S₁ = √(16 + 9 − 8 − 4) = √13.

The other 25 questions

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6. Q6. With respect to the circle x² + y² − 4x + 2y − 4 = 0, the point (1, 1) lies:

   • A. outside the circle
   • B. on the circle
   • C. at the centre
   • D. inside the circle

7. Q7. The focus of the parabola y² = −12x is:

   • A. (−12, 0)
   • B. (−3, 0)
   • C. (3, 0)
   • D. (0, −3)

8. Q8. The directrix of the parabola x² = 6y is:

   • A. x = −3/2
   • B. y = −3/2
   • C. y = −6
   • D. y = 3/2

9. Q9. The vertex of the parabola y² − 4y − 8x + 12 = 0 is:

   • A. (−1, 2)
   • B. (1, −2)
   • C. (2, 1)
   • D. (1, 2)

10. Q10. The length of the latus rectum of y² = 20x is:

    • A. 5
    • B. 20
    • C. 40
    • D. 10

11. Q11. A parabola has its vertex at the origin, axis along the x-axis, and passes through (2, 4). Its equation is:

    • A. y² = 4x
    • B. y² = 8x
    • C. x² = 8y
    • D. y² = 16x

12. Q12. The foci of the ellipse x²/9 + y²/25 = 1 are:

    • A. (0, ±√34)
    • B. (0, ±4)
    • C. (±3, 0)
    • D. (±4, 0)

13. Q13. The eccentricity of the ellipse 9x² + 16y² = 144 is:

    • A. √7/3
    • B. 3/4
    • C. 5/4
    • D. √7/4

14. Q14. An ellipse centred at the origin has its major axis along the x-axis, semi-major axis 4 and eccentricity 1/2. Its equation is:

    • A. x²/16 + y²/8 = 1
    • B. x²/16 + y²/4 = 1
    • C. x²/16 + y²/12 = 1
    • D. x²/4 + y²/16 = 1

15. Q15. For any point on the ellipse x²/36 + y²/20 = 1, the sum of its distances from the two foci is:

    • A. 8
    • B. 20
    • C. 6
    • D. 12

16. Q16. The eccentricity of the hyperbola x²/25 − y²/144 = 1 is:

    • A. 13/12
    • B. 5/13
    • C. 13/5
    • D. 12/5

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

    • A. (±3, 0)
    • B. (0, ±4)
    • C. (0, ±5)
    • D. (0, ±3)

18. Q18. The length of the latus rectum of the hyperbola x²/4 − y²/12 = 1 is:

    • A. 12
    • B. 3
    • C. 24
    • D. 6

19. Q19. The hyperbola with foci (±10, 0) and vertices (±6, 0) is:

    • A. x²/36 − y²/100 = 1
    • B. x²/64 − y²/36 = 1
    • C. x²/36 + y²/64 = 1
    • D. x²/36 − y²/64 = 1

20. Q20. The equation x² − 4x − y + 1 = 0 represents:

    • A. a parabola
    • B. a circle
    • C. an ellipse
    • D. a hyperbola

21. Q21. The equation 3x² + 3y² − 6x + 12y = 0 represents a circle with:

    • A. centre (−1, 2) and radius √5
    • B. centre (1, −2) and radius 5
    • C. centre (3, −6) and radius 5
    • D. centre (1, −2) and radius √5

22. Q22. A conic whose eccentricity e satisfies 0 < e < 1 is:

    • A. a hyperbola
    • B. a parabola
    • C. an ellipse
    • D. a pair of lines

23. Q23. For what values of k does the line 3x + 4y = k touch the circle x² + y² = 4?

    • A. k = ±20
    • B. k = ±2
    • C. k = ±10
    • D. k = ±5

24. Q24. The tangent to the parabola y² = 12x having slope 3 is:

    • A. y = 3x + 1
    • B. y = 3x + 4
    • C. y = 3x − 1
    • D. y = 3x + 3

25. Q25. The circles x² + y² = 9 and (x − 5)² + y² = 4:

    • A. do not meet at all
    • B. touch each other externally
    • C. intersect in two points
    • D. touch each other internally

26. Q26. The curve x = 3 sec θ, y = 2 tan θ is:

    • A. the hyperbola x²/9 − y²/4 = 1
    • B. the circle x² + y² = 13
    • C. the hyperbola x²/4 − y²/9 = 1
    • D. the ellipse x²/9 + y²/4 = 1

27. Q27. The circle through the three points (0, 0), (4, 0) and (0, 6) is:

    • A. x² + y² − 4x − 6y = 0
    • B. x² + y² − 6x − 4y = 0
    • C. x² + y² + 4x + 6y = 0
    • D. x² + y² − 2x − 3y = 0

28. Q28. A point P on the parabola y² = 8x has x-coordinate 3. Its distance from the focus is:

    • A. 5
    • B. 2√6
    • C. 7
    • D. 3

29. Q29. The directrices of the ellipse x²/25 + y²/9 = 1 are:

    • A. x = ±5/4
    • B. x = ±16/5
    • C. x = ±25/4
    • D. x = ±4

30. Q30. The circle with centre (3, 4) that touches the x-axis is:

    • A. (x − 3)² + (y − 4)² = 25
    • B. (x − 3)² + (y − 4)² = 16
    • C. (x − 3)² + (y − 4)² = 9
    • D. (x + 3)² + (y + 4)² = 16