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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Conic Sections (Set 3)

Take the ECAT Mathematics — Conic Sections (Set 3): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30…

What this test covers

Sample questions with answers

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

  1. Q1. The circle with centre (1, −2) and radius 3 has the equation:

    • A. x² + y² − 2x + 4y + 4 = 0
    • B. x² + y² − 2x + 4y − 9 = 0
    • C. x² + y² − 2x + 4y − 4 = 0 ✓
    • D. x² + y² + 2x − 4y − 4 = 0

    Answer: C. x² + y² − 2x + 4y − 4 = 0

    Explanation: (x − 1)² + (y + 2)² = 9 expands to x² + y² − 2x + 4y + 5 − 9 = 0, i.e. x² + y² − 2x + 4y − 4 = 0.

  2. Q2. The radius of the circle 2x² + 2y² − 8x + 12y + 2 = 0 is:

    • A. 2√3 ✓
    • B. √3
    • C. √14
    • D. √13

    Answer: A. 2√3

    Explanation: Divide by 2: x² + y² − 4x + 6y + 1 = 0, so g = −2, f = 3, c = 1. r = √(g² + f² − c) = √12 = 2√3. (Using the undivided equation gives a wrong radius.)

  3. Q3. The circle passing through (0, 0), (2, 0) and (0, −4) is:

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

    Answer: D. x² + y² − 2x + 4y = 0

    Explanation: Through the origin: x² + y² + 2gx + 2fy = 0. (2, 0) gives 4 + 4g = 0, g = −1. (0, −4) gives 16 − 8f = 0, f = 2. So x² + y² − 2x + 4y = 0.

  4. Q4. The circle of radius 5 lying in the first quadrant and touching both coordinate axes has equation:

    • A. x² + y² = 25
    • B. (x − 5)² + (y − 5)² = 25 ✓
    • C. (x − 5)² + (y − 5)² = 5
    • D. (x + 5)² + (y + 5)² = 25

    Answer: B. (x − 5)² + (y − 5)² = 25

    Explanation: Touching both axes means the centre is at distance 5 from each: (5, 5). The equation is (x − 5)² + (y − 5)² = 25.

  5. Q5. The line 3x + 4y = 10 cuts the circle x² + y² = 9 in a chord of length:

    • A. 4
    • B. 2√13
    • C. √5
    • D. 2√5 ✓

    Answer: D. 2√5

    Explanation: Distance from the centre (0, 0) to the line is 10/5 = 2. Half-chord = √(9 − 4) = √5, so the chord is 2√5.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

  1. Q6. The length of the tangent from the point (7, 1) to the circle x² + y² − 4x − 6y + 9 = 0 is:

    • A. 7
    • B. √5
    • C. 25
    • D. 5

  2. Q7. The tangent to the circle x² + y² = 20 at the point (4, 2) is:

    • A. 4x + 2y = 10
    • B. x + 2y = 10
    • C. 2x + y = 10
    • D. 2x − y = 6

  3. Q8. The directrix of the parabola y² = −8x is:

    • A. y = 2
    • B. x = 2
    • C. x = 8
    • D. x = −2

  4. Q9. The focus of the parabola x² = −16y is:

    • A. (0, −16)
    • B. (0, −4)
    • C. (0, 4)
    • D. (−4, 0)

  5. Q10. The parabola with vertex (1, 2) and focus (3, 2) has equation:

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

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

    • A. (1/4, 0)
    • B. (0, 4)
    • C. (0, 1/4)
    • D. (0, 1)

  7. Q12. The ends of the latus rectum of the parabola y² = 12x are:

    • A. (12, 6) and (12, −6)
    • B. (3, 12) and (3, −12)
    • C. (6, 3) and (−6, 3)
    • D. (3, 6) and (3, −6)

  8. Q13. The parabola y² = 4ax passes through the point (4, 8). Its focus is:

    • A. (2, 0)
    • B. (16, 0)
    • C. (0, 4)
    • D. (4, 0)

  9. Q14. The eccentricity of the ellipse x²/49 + y²/24 = 1 is:

    • A. 5/√24
    • B. 25/49
    • C. 5/7
    • D. √24/7

  10. Q15. The length of the minor axis of the ellipse 16x² + 25y² = 400 is:

    • A. 5
    • B. 8
    • C. 16
    • D. 10

  11. Q16. The length of the latus rectum of the ellipse 9x² + 25y² = 225 is:

    • A. 9/5
    • B. 18/5
    • C. 6
    • D. 10/3

  12. Q17. The ellipse with vertices (0, ±5) and foci (0, ±3) has equation:

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

  13. Q18. An ellipse has foci (±2, 0) and eccentricity 1/2. Its equation is:

    • A. x²/4 + y²/3 = 1
    • B. x²/12 + y²/16 = 1
    • C. x²/16 + y²/12 = 1
    • D. x²/16 + y²/4 = 1

  14. Q19. For any point P on the ellipse x²/100 + y²/36 = 1, the sum of the distances of P from the two foci is:

    • A. 20
    • B. 10
    • C. 16
    • D. 8

  15. Q20. The asymptotes of the hyperbola 9x² − 16y² = 144 are:

    • A. y = ±(3/4)x
    • B. y = ±(9/16)x
    • C. y = ±(3/16)x
    • D. y = ±(4/3)x

  16. Q21. The eccentricity of the hyperbola 16y² − 9x² = 144 is:

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

  17. Q22. The hyperbola with foci (±√13, 0) and vertices (±2, 0) has equation:

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

  18. Q23. The parabola y² = 16x has a tangent with slope 1. Its equation is:

    • A. y = x + 1
    • B. y = x + 16
    • C. y = x + 4
    • D. y = x − 4

  19. Q24. The equation x² − 4y² + 8y − 4 = 0 represents:

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

  20. Q25. The equation 3x² + 3y² − 6x + 12y + 15 = 0 represents:

    • A. no real locus
    • B. a circle of radius 2
    • C. a single point
    • D. a circle of radius 1

  21. Q26. A circle has the points (−1, 2) and (3, −4) as the ends of a diameter. Its equation is:

    • A. x² + y² − 2x + 2y + 11 = 0
    • B. x² + y² − 2x + 2y − 13 = 0
    • C. x² + y² − 2x + 2y − 11 = 0
    • D. x² + y² + 2x − 2y − 11 = 0

  22. Q27. The line x + y = 7 meets the circle x² + y² = 25 at the points:

    • A. (0, 7) and (7, 0)
    • B. (3, 4) and (−4, −3)
    • C. (5, 2) and (2, 5)
    • D. (3, 4) and (4, 3)

  23. Q28. The parametric equations x = 2 + 3cos θ, y = −1 + 3sin θ represent:

    • A. a circle with centre (−2, 1) and radius 3
    • B. a circle with centre (2, −1) and radius 9
    • C. an ellipse centred at (2, −1)
    • D. a circle with centre (2, −1) and radius 3

  24. Q29. The circle with centre (3, 4) that passes through the origin has equation:

    • A. x² + y² + 6x + 8y = 0
    • B. x² + y² − 6x − 8y = 0
    • C. x² + y² − 6x − 8y − 25 = 0
    • D. x² + y² − 6x − 8y + 25 = 0

  25. Q30. The foci of the ellipse 9x² + 4y² = 36 are:

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