ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Conic Sections (Set 4): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. The radius of the circle x² + y² − 6x + 8y = 0 is
Answer: D. 5
Explanation: Completing squares: (x − 3)² + (y + 4)² = 25, so the radius is 5 (the centre is (3, −4)).
Q2. The focus of the parabola x² = 20y is
Answer: B. (0, 5)
Explanation: x² = 4ay with 4a = 20 gives a = 5, so the focus is (0, 5).
Q3. An ellipse has semi-major axis 10 and semi-minor axis 6. Its eccentricity is
Answer: A. 4/5
Explanation: e = √(1 − b²/a²) = √(1 − 36/100) = √(64/100) = 4/5.
Q4. The foci of the conic 5y² − 4x² = 20 are
Answer: D. (0, ±3)
Explanation: Dividing by 20: y²/4 − x²/5 = 1. The y² term is positive, so the transverse axis is vertical, with c² = 4 + 5 = 9. The foci are (0, ±3).
Q5. The length of the latus rectum of the parabola y² = 8x is
Answer: A. 8
Explanation: For y² = 4ax the latus rectum is 4a. Here 4a = 8.
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Q6. The tangent to the circle x² + y² = 25 at the point (3, 4) is
Q7. The circle through (0, 0), (2, 0) and (0, 4) has radius
Q8. The directrix of the parabola x² = −16y is
Q9. The length of the latus rectum of the ellipse 9x² + 16y² = 144 is
Q10. The eccentricity of the hyperbola 25x² − 16y² = 400 is
Q11. The line y = 2x + c is tangent to the parabola y² = 8x when c equals
Q12. The line y = x + c touches the circle x² + y² = 8 for a positive c equal to
Q13. The distance between the foci of the ellipse x²/49 + y²/24 = 1 is
Q14. The equation x² + 2xy + y² + 3x + 2 = 0 represents
Q15. The circle x² + y² − 6x − 4y + k = 0 touches the x-axis when k equals
Q16. The circle on the line segment joining (1, 2) and (5, 6) as diameter has equation
Q17. The parabola with vertex at the origin and focus (0, −2) has equation
Q18. The ellipse with foci (±3, 0) and major axis of length 10 has equation
Q19. For the ellipse 4x² + 9y² − 16x + 18y − 11 = 0, the foci are
Q20. The length of the tangent from (5, 4) to the circle x² + y² − 2x − 4y − 4 = 0 is
Q21. The length of the common chord of the circles x² + y² = 25 and (x − 6)² + y² = 25 is
Q22. The tangent to the parabola y² = 16x at the point (1, 4) is
Q23. The locus of a point P such that |PF₁ − PF₂| = 6, where F₁ = (5, 0) and F₂ = (−5, 0), is
Q24. The focal chord of y² = 4x passing through the point (4, 4) has length
Q25. The line y = x + k is tangent to the ellipse x²/9 + y²/4 = 1 for a positive k equal to
Q26. A hyperbola x²/a² − y²/b² = 1 has asymptotes y = ±x/2. Its eccentricity is
Q27. The circles x² + y² = 1 and (x − 3)² + y² = 4 have how many common tangents?
Q28. A parabola with vertex at the origin and the x-axis as axis passes through (3, 6). Its latus rectum has length
Q29. An ellipse has its latus rectum equal to the distance between its foci. Its eccentricity is
Q30. Tangents are drawn from (5, 0) to the circle x² + y² = 9. The area of the quadrilateral formed by the centre, the external point and the two points of contact is