ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Conic Sections (Set 6)

Take the ECAT Mathematics — Conic Sections (Set 6): 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 circle through the points (0, 0), (4, 0) and (0, 6) has radius:

   • A. 5
   • B. √26
   • C. √10
   • D. √13 ✓

   Answer: D. √13

   Explanation: The right angle at the origin means the hypotenuse from (4, 0) to (0, 6) is a diameter of length √(16 + 36) = 2√13, so the radius is √13.

2. Q2. Circles tangent to both coordinate axes in the first quadrant pass through the point (2, 1). The largest such radius is:

   • A. 3
   • B. 1
   • C. 4
   • D. 5 ✓

   Answer: D. 5

   Explanation: Centre (r, r): (r − 2)² + (r − 1)² = r² ⇒ r² − 6r + 5 = 0 ⇒ r = 1 or 5. The larger is 5.

3. Q3. The line x + y = 2 cuts the circle x² + y² = 10 in a chord of length:

   • A. 4√2 ✓
   • B. 2√2
   • C. 8
   • D. 2√10

   Answer: A. 4√2

   Explanation: Distance from the origin to the line = 2/√2 = √2. Chord = 2√(r² − d²) = 2√(10 − 2) = 2√8 = 4√2.

4. Q4. For which positive value of c is the line y = x + c tangent to the circle x² + y² = 8?

   • A. 8
   • B. 2
   • C. 4 ✓
   • D. 2√2

   Answer: C. 4

   Explanation: Distance from the centre (0, 0) to x − y + c = 0 is |c|/√2 and must equal the radius 2√2. So |c| = 4, giving c = 4.

5. Q5. The circles x² + y² = 9 and (x − 4)² + y² = 9 intersect. The length of their common chord is:

   • A. 2√5 ✓
   • B. 4
   • C. 6
   • D. √5

   Answer: A. 2√5

   Explanation: Subtracting the equations gives the common chord x = 2. Then y² = 9 − 4 = 5, so y = ±√5 and the chord has length 2√5.

The other 25 questions

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6. Q6. A point on the parabola y² = 12x is at a distance 7 from the focus. Its ordinate (y-coordinate) can be:

   • A. 2√7
   • B. 4√3
   • C. 6
   • D. 3√5

7. Q7. The equation of the tangent to the parabola x² = 8y at the point (4, 2) is:

   • A. 2x − y − 6 = 0
   • B. x − 2y = 0
   • C. x + y − 6 = 0
   • D. x − y − 2 = 0

8. Q8. The focus of the parabola y² − 4y − 8x + 12 = 0 is:

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

9. Q9. The parabola with focus (3, −1) and directrix y = 3 has the equation:

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

10. Q10. The length of the latus rectum of the ellipse 16x² + 25y² = 400 is:

    • A. 16/5
    • B. 8
    • C. 9/5
    • D. 32/5

11. Q11. An ellipse has foci (±3, 0) and passes through (0, 4). The sum of the focal distances of any point on it is:

    • A. 10
    • B. 6
    • C. 25
    • D. 8

12. Q12. The largest area of a rectangle with sides parallel to the axes inscribed in the ellipse x²/9 + y²/4 = 1 is:

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

13. Q13. The ellipse x²/16 + y²/7 = 1 has foci F₁(−3, 0) and F₂(3, 0). If P lies on it with PF₁ = 5, then PF₂ equals:

    • A. 5
    • B. 3
    • C. 8
    • D. 4

14. Q14. The hyperbola x²/9 − y²/16 = 1 has eccentricity 5/3. The eccentricity of its conjugate hyperbola is:

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

15. Q15. The distance between the vertices of the rectangular hyperbola xy = 18 is:

    • A. 6
    • B. 6√2
    • C. 18
    • D. 12

16. Q16. The conic 4x² + 9y² − 8x + 36y + 4 = 0 has its centre at:

    • A. (−1, 2)
    • B. (2, −4)
    • C. (1, 2)
    • D. (1, −2)

17. Q17. Which of the following equations represents a parabola?

    • A. x² + 2xy + y² + x = 0
    • B. x² + 3xy + y² = 1
    • C. x² − xy + y² − 3 = 0
    • D. x² + xy + y² = 1

18. Q18. The tangent to x² + y² = 25 at (3, 4) forms a triangle with the coordinate axes of area:

    • A. 25/2
    • B. 625/12
    • C. 50/3
    • D. 625/24

19. Q19. The equation x² + y² + 4x − 6y + k = 0 represents a real circle (with positive radius) for:

    • A. k ≤ 13
    • B. k < 13
    • C. k > 13
    • D. k < 0

20. Q20. The tangents drawn from the point (−2, 0) to the parabola y² = 8x are inclined to each other at:

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

21. Q21. The area of the circle x² + y² − 6x + 8y = 0 is:

    • A. 5π
    • B. 50π
    • C. 10π
    • D. 25π

22. Q22. The ends of the latus rectum of the parabola x² = −12y are:

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

23. Q23. The equation of the circle having the points (1, 2) and (5, 6) as ends of a diameter is:

    • A. x² + y² − 3x − 4y + 17 = 0
    • B. x² + y² − 6x − 8y + 7 = 0
    • C. x² + y² + 6x + 8y + 17 = 0
    • D. x² + y² − 6x − 8y + 17 = 0

24. Q24. An ellipse has a latus rectum equal to half its minor axis. Its eccentricity is:

    • A. 3/4
    • B. 1/2
    • C. √2/2
    • D. √3/2

25. Q25. The vertex of the parabola y = x² − 6x + 11 is the point:

    • A. (−3, 2)
    • B. (3, 2)
    • C. (6, 11)
    • D. (3, 11)

26. Q26. Two parabolas y² = 4x and x² = 4y intersect at the origin and at the point:

    • A. (1, 1)
    • B. (2, 2)
    • C. (4, −4)
    • D. (4, 4)

27. Q27. The common chord (radical axis) of the circles x² + y² − 4 = 0 and x² + y² − 2x − 6y + 6 = 0 is:

    • A. x + 3y + 5 = 0
    • B. x + 3y − 5 = 0
    • C. x + y − 5 = 0
    • D. x − 3y + 5 = 0

28. Q28. A point moves so that its distance from (3, 0) is 3/2 times its distance from the line 3x = 4. Its locus is:

    • A. x²/4 − y²/5 = 1
    • B. x²/5 − y²/4 = 1
    • C. x²/4 + y²/5 = 1
    • D. x²/9 − y²/4 = 1

29. Q29. The circles x² + y² = 16 and (x − 1)² + y² = 9 have how many common tangents?

    • A. 1
    • B. 0
    • C. 3
    • D. 2

30. Q30. The ellipse x²/9 + y²/5 = 1 has foci (±2, 0). The distance of the point (2, 5/3) on the ellipse from the focus (−2, 0) is:

    • A. 4
    • B. 5/3
    • C. 11/3
    • D. 13/3