Online Test — Conic Sections
10 Questions • 20 min • Chapter MCQ
20:00
Question 1 of 10
easy
Find the equation of the parabola with focus (3, 0) and directrix x = −3.
y² = -12x
x² = 12y
y² = 12x
y² = 3x
Explanation: For a standard parabola opening to the right, the focus is (a, 0) and directrix is x = -a. Here a = 3. The equation is y² = 4ax, which becomes y² = 12x.
Question 2 of 10
medium
The focus of y² + 8x = 0 is:
(-2, 0)
(2, 0)
(0, -2)
(0, 2)
Explanation: y² = -8x gives a = -2. Focus = (-2, 0).
Question 3 of 10
medium
Determine the vertex of the parabola x = y² - 4y + 2.
(2, -2)
(-2, 2)
(2, 2)
(-2, -2)
Explanation: Complete the square for y: x = (y² - 4y + 4) - 2 → x = (y - 2)² - 2 → x + 2 = (y - 2)². The vertex (h, k) is (-2, 2).
Question 4 of 10
medium
The headlight reflector of an Indian Tata truck is a paraboloid 20 cm wide and 5 cm deep. At what distance from the vertex should the bulb (focus) be placed?
2 cm
4 cm
5 cm
10 cm
Explanation: Let the vertex be (0,0) and the axis be the x-axis. The equation is y² = 4ax. The points on the rim are (5, 10) and (5, -10). Substitute (5, 10): 10² = 4a(5) → 100 = 20a → a = 5 cm.
Question 5 of 10
easy
If the directrix of a parabola is the x-axis and its focus is (0, 10), where is the vertex?
(0, 0)
(0, 5)
(5, 0)
(0, 10)
Explanation: The vertex lies exactly halfway between the focus (0, 10) and the directrix (y = 0). Midpoint is (0, (10+0)/2) = (0, 5).
Question 6 of 10
medium
Find the radical axis of the circles x² + y² − 2x = 0 and x² + y² − 4y = 0.
x − 2y = 0
x + 2y = 0
2x − y = 0
2x + y = 0
Explanation: The radical axis of two circles S1 = 0 and S2 = 0 is S1 − S2 = 0. Subtracting gives (x² + y² − 2x) − (x² + y² − 4y) = 0, which is −2x + 4y = 0, or x − 2y = 0.
Question 7 of 10
easy
What are the coordinates of the focus for the parabola y² = -16x?
(4, 0)
(-4, 0)
(0, -4)
(0, 4)
Explanation: Comparing y² = -16x with y² = -4ax gives 4a = 16, so a = 4. Since it opens to the left, the focus is (-a, 0), which is (-4, 0).
Question 8 of 10
easy
Which equation represents an ellipse?
x²/16 + y²/9 = 1
x² − y² = 1
y² = 4x
x² + y² + 1 = 0
Explanation: Sum of positive squared terms equals 1.
Question 9 of 10
medium
The foci of x²/16 + y²/64 = 1 are:
(±4√3,0)
(0,±4√3)
(0,±8)
(±8,0)
Explanation: c² = 64 − 16 = 48, so c = 4√3.
Question 10 of 10
medium
If a point P(x₁, y₁) lies strictly inside the ellipse x²/a² + y²/b² = 1, then the value of x₁²/a² + y₁²/b² - 1 is:
Strictly less than 0
Strictly greater than 0
Equal to 0
Undefined
Explanation: For interior points, the expression S₁ = x₁²/a² + y₁²/b² - 1 is negative (< 0).