ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Conic Sections (Set 5)

Take the ECAT Mathematics — Conic Sections (Set 5): 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 radical axis of the circles x² + y² − 4x = 0 and x² + y² − 6y = 0 is:

   • A. 2x + 3y = 0
   • B. 4x − 6y = 1
   • C. 2x − 3y = 0 ✓
   • D. 3x − 2y = 0

   Answer: C. 2x − 3y = 0

   Explanation: Subtracting the equations: (−4x) − (−6y) = 0, i.e. −4x + 6y = 0, or 2x − 3y = 0.

2. Q2. Circles in the first quadrant touch both coordinate axes and pass through (2, 1). The sum of their radii is:

   • A. 5
   • B. 4
   • C. 3
   • D. 6 ✓

   Answer: D. 6

   Explanation: Centre (r, r): (2 − r)² + (1 − r)² = r² gives r² − 6r + 5 = 0, so r = 1 or 5; the sum is 6.

3. Q3. A circle passes through (1, 2) and (3, 4) and has its centre on the y-axis. Its centre is:

   • A. (0, 5) ✓
   • B. (0, 4)
   • C. (0, 10)
   • D. (0, 3)

   Answer: A. (0, 5)

   Explanation: Centre (0, k): 1 + (2 − k)² = 9 + (4 − k)² gives −4k + 5 = −8k + 25, so k = 5 (radius² = 10).

4. Q4. The chord of contact of the tangents drawn from (4, 3) to the circle x² + y² = 16 is the line:

   • A. 3x + 4y = 16
   • B. 4x − 3y = 16
   • C. 4x + 3y = 16 ✓
   • D. 4x + 3y = 25

   Answer: C. 4x + 3y = 16

   Explanation: For an external point (x₁, y₁) the chord of contact is xx₁ + yy₁ = r²: 4x + 3y = 16. (The point is outside: 16 + 9 > 16.)

5. Q5. The equation of the normal to y² = 4x at the point (4, 4) is:

   • A. x − 2y = −4
   • B. x + 2y = 12
   • C. 2x + y = 12 ✓
   • D. 2x − y = 4

   Answer: C. 2x + y = 12

   Explanation: dy/dx = 2/y = 1/2 at (4, 4), so the normal has slope −2: y − 4 = −2(x − 4), i.e. 2x + y = 12.

The other 25 questions

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6. Q6. A point P on the parabola x² = 12y is at distance 8 from the focus. The y-coordinate of P is:

   • A. 8
   • B. 3
   • C. 11
   • D. 5

7. Q7. The parabola with focus (3, 1) and directrix x = −1 has equation:

   • A. (x − 1)² = 8(y − 1)
   • B. (y − 1)² = 8(x − 1)
   • C. (y − 1)² = 4(x − 1)
   • D. y² = 8x

8. Q8. An ellipse has foci (±3, 0) and passes through (0, 4). Its equation is:

   • A. x²/16 + y²/7 = 1
   • B. x²/25 + y²/16 = 1
   • C. x²/16 + y²/25 = 1
   • D. x²/25 + y²/9 = 1

9. Q9. An ellipse centred at the origin has foci (±4, 0) and directrices x = ±25/4. Its equation is:

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

10. Q10. The end of the latus rectum of x²/25 + y²/9 = 1 that lies in the first quadrant is:

    • A. (4, 9/5)
    • B. (4, 18/5)
    • C. (4, 3/5)
    • D. (3, 16/5)

11. Q11. For the ellipse x²/25 + y²/9 = 1, the distance of the focus (4, 0) from the point of the ellipse with abscissa 5/2 is:

    • A. 7
    • B. 4
    • C. 3
    • D. 2

12. Q12. The eccentricity of the conjugate of the hyperbola x²/4 − y²/5 = 1 is:

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

13. Q13. A hyperbola has asymptotes y = ±(3/4)x and vertices (±8, 0). Its equation is:

    • A. x²/16 − y²/9 = 1
    • B. x²/64 − y²/48 = 1
    • C. x²/64 − y²/36 = 1
    • D. x²/64 − y²/9 = 1

14. Q14. The conic 9x² − 4y² − 18x − 16y − 43 = 0 has its centre at:

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

15. Q15. A parabola with vertical axis has vertex (2, −1) and passes through (4, 0). Its latus rectum has length:

    • A. 8
    • B. 1
    • C. 4
    • D. 2

16. Q16. A point moves so that its distance from (2, 0) is half its distance from the line x = 8. Its locus is:

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

17. Q17. The line y = 2x + c touches the hyperbola x²/9 − y²/16 = 1 when c equals:

    • A. ±√7
    • B. ±4
    • C. ±4√5
    • D. ±2√5

18. Q18. The tangent to the circle x² + y² − 4x + 2y − 20 = 0 at the point (5, 3) is:

    • A. 3x + 4y = 27
    • B. 3x − 4y = 3
    • C. 3x + 4y = 15
    • D. 4x + 3y = 29

19. Q19. The circles (x − 1)² + y² = 9 and (x − 3)² + y² = 1 have how many common tangents?

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

20. Q20. The locus of the midpoints of chords of x² + y² = 16 drawn through the point (4, 0) is:

    • A. x² + y² + 4x = 0
    • B. x² + y² = 4
    • C. x² + y² − 8x = 0
    • D. x² + y² − 4x = 0

21. Q21. If the minor axis of an ellipse equals the distance between its foci, its eccentricity is:

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

22. Q22. The latus rectum of a hyperbola equals its conjugate axis. Its eccentricity is:

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

23. Q23. The equation x² + y² − 4x + 6y + 13 = 0 represents:

    • A. a single point
    • B. a circle of radius 2
    • C. a circle of radius 3
    • D. no real locus

24. Q24. The line x + y = 8 cuts the circle x² + y² = 50. The midpoint of the chord is:

    • A. (4, 4)
    • B. (3, 5)
    • C. (8, 0)
    • D. (5, 3)

25. Q25. The parametric equations x = 1 + 2 cos θ, y = −2 + 3 sin θ represent an ellipse with centre:

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

26. Q26. The number of points common to the circle x² + y² = 4 and the ellipse x²/9 + y²/4 = 1 is:

    • A. 2
    • B. 4
    • C. 0
    • D. 1

27. Q27. The length of the chord of the parabola y² = 4x cut off by the line y = x − 1 is:

    • A. 4√2
    • B. 8
    • C. 6
    • D. 4

28. Q28. The distance between the directrices of the ellipse x²/16 + y²/7 = 1 is:

    • A. 24
    • B. 16/3
    • C. 8/3
    • D. 32/3

29. Q29. The vertex and the ends of the latus rectum of the parabola y² = 8x form a triangle of area:

    • A. 12
    • B. 16
    • C. 4
    • D. 8

30. Q30. The line y = x + 1 meets the parabola y² = 4x at:

    • A. two distinct points
    • B. no point
    • C. exactly one point, but it is not a tangent
    • D. exactly one point (it is a tangent)