Conic Sections (Circles, Ellipses, Parabolas, Hyperbolas) — Practice

Question 1

Find the equation of a circle with centre (2, −3) and radius 4.

Accepted Answer

(x−2)² + (y+3)² = 16

Question 2

Find the focus, directrix, and length of latus rectum of y² = 12x.

Accepted Answer

Comparing y²=4ax: 4a=12 → a=3. Focus = (3, 0). Directrix: x = −3. Length of latus rectum = 4a = 12.

Question 3

Find the equation of the ellipse with foci (±3, 0) and a = 5.

Accepted Answer

c=3, a=5 → b²=a²−c²=25−9=16. Equation: x²/25 + y²/16 = 1.

Question 4

For the hyperbola x²/9 − y²/16 = 1, find eccentricity, foci, and equations of asymptotes.

Accepted Answer

a²=9, b²=16 → c²=a²+b²=25 → c=5. e=c/a=5/3. Foci: (±5, 0). Asymptotes: y=±(b/a)x = ±(4/3)x.

Practice — Conic Sections (Circles, Ellipses, Parabolas, Hyperbolas)