For Q5 to Q10, find the foci, the vertices, the major and minor axes, the eccentricity and the latus rectum.
Q5. x2/16 + y2/9 = 1 (NCERT Exercise 10.3, Q3).
a = 4, b = 3, c = √7: foci (±√7, 0), vertices (±4, 0), axes 8 and 6, e = √7/4, latus rectum 9/2.
Q6. x2/49 + y2/36 = 1 (NCERT Exercise 10.3, Q5).
a = 7, b = 6, c = √13: foci (±√13, 0), vertices (±7, 0), axes 14 and 12, e = √13/7, latus rectum 72/7.
Q7. x2/100 + y2/400 = 1 (NCERT Exercise 10.3, Q6).
Major axis along y: a = 20, b = 10, c = √300 = 10√3. Foci (0, ±10√3), vertices (0, ±20), axes 40 and 20, e = √3/2, latus rectum 10.
Q8. 36x2 + 4y2 = 144 (NCERT Exercise 10.3, Q7).
x2/4 + y2/36 = 1: a = 6 along y, b = 2, c = √32 = 4√2. Foci (0, ±4√2), vertices (0, ±6), axes 12 and 4, e = 2√2/3, latus rectum 4/3.
Q9. 16x2 + y2 = 16 (NCERT Exercise 10.3, Q8).
x2/1 + y2/16 = 1: a = 4 along y, b = 1, c = √15. Foci (0, ±√15), vertices (0, ±4), axes 8 and 2, e = √15/4, latus rectum 1/2.
Q10. 4x2 + 9y2 = 36 (NCERT Exercise 10.3, Q9).
x2/9 + y2/4 = 1: a = 3, b = 2, c = √5. Foci (±√5, 0), vertices (±3, 0), axes 6 and 4, e = √5/3, latus rectum 8/3.
Q11. Find the ellipse with vertices (0, ±13) and foci (0, ±5) (NCERT Exercise 10.3, Q11).
b2 = 169 − 25 = 144: x2/144 + y2/169 = 1.
Q12. Find the ellipse with vertices (±6, 0) and foci (±4, 0) (NCERT Exercise 10.3, Q12).
b2 = 36 − 16 = 20: x2/36 + y2/20 = 1.
Q13. Find the ellipse with ends of the major axis (±3, 0) and of the minor axis (0, ±2) (NCERT Exercise 10.3, Q13).
a = 3, b = 2: x2/9 + y2/4 = 1.
Q14. Find the ellipse with ends of the major axis (0, ±√5) and of the minor axis (±1, 0) (NCERT Exercise 10.3, Q14).
a = √5 along y, b = 1: x2/1 + y2/5 = 1.
Q15. Find the ellipse with major axis 26 and foci (±5, 0) (NCERT Exercise 10.3, Q15).
a = 13, b2 = 169 − 25 = 144: x2/169 + y2/144 = 1.
Q16. Find the ellipse with foci (±3, 0) and a = 4 (NCERT Exercise 10.3, Q17).
b2 = 16 − 9 = 7: x2/16 + y2/7 = 1.
Q17. Find the ellipse with b = 3, c = 4, centre at the origin and foci on the x-axis (NCERT Exercise 10.3, Q18).
a2 = 9 + 16 = 25: x2/25 + y2/9 = 1.
Q18. Find the ellipse with centre (0, 0), major axis on the y-axis, through (3, 2) and (1, 6) (NCERT Exercise 10.3, Q19).
x2/b2 + y2/a2 = 1 with p = 1/b2, q = 1/a2: 9p + 4q = 1 and p + 36q = 1. Then q = 1/40 and p = 1/10:
x2/10 + y2/40 = 1. Here 40 > 10, so the major axis is indeed along y.
Q19. Find the ellipse centred at the origin with major axis on the x-axis, through (4, 3) and (6, 2) (NCERT Exercise 10.3, Q20).
With p = 1/a2, q = 1/b2: 16p + 9q = 1 and 36p + 4q = 1. Then p = 1/52 and q = 1/13:
x2/52 + y2/13 = 1.
Q20. A rod AB of length 15 cm rests with A on the x-axis and B on the y-axis. P is the point of the rod with AP = 6 cm. Show that the locus of P is an ellipse (NCERT Miscellaneous Example 19, pages 203–204).
Parameterize the endpoints as A = (15 cos θ, 0), B = (0, 15 sin θ), which ensures AB = 15. Since AP : PB = 6 : 9, the section formula gives P = (9 cos θ, 6 sin θ). Since cos2 θ + sin2 θ = 1, x2/81 + y2/36 = 1. With endpoints allowed on the full axes, this is the whole ellipse; if they are restricted to the positive axes, only its first-quadrant arc is traced.
Q21. An arch is a semi-ellipse 8 m wide and 2 m high at the centre. Find its height 1.5 m from one end (NCERT Miscellaneous Exercise, Q4).
x2/16 + y2/4 = 1. At 1.5 m from an end, x = 2.5, so y2 = 4(1 − 6.25/16) = 2.4375 and y ≈ 1.56 m.
Q22. A rod of length 12 cm moves with its ends on the coordinate axes. Find the locus of the point P of the rod 3 cm from the end on the x-axis (NCERT Miscellaneous Exercise, Q5).
As in Q20, with AP = 3 and PB = 9: P = (9 cos θ, 3 sin θ), so x2/81 + y2/9 = 1.
Q23. A man running a racecourse notes that the sum of his distances from two flag posts is always 10 m, and the posts are 8 m apart. Find the equation of his path (NCERT Miscellaneous Exercise, Q7).
An ellipse with 2a = 10 and 2c = 8: a = 5, c = 4, b = 3. With the posts at (±4, 0), x2/25 + y2/9 = 1.