Parabola — Chapter Test | Class 11 IMO

Q1

A parabola has directrix y = -6 and vertex at origin. Its focus is:

Focus is (0, a) where a = 6.

Q2

If the focus is (6, 0), then the directrix is:

For focus (a,0), directrix is x = -a.

Q3

Ravi hits a cricket ball whose path is modeled by y = 2x - 0.1x² (x and y in meters). Find the maximum height reached by the ball.

The path is y = -0.1(x² - 20x). Completing the square gives y = -0.1(x - 10)² + 10. The vertex is at (10, 10). The maximum height is the y-coordinate of the vertex, which is 10 m.

Q4

The focal distance of a point P(x, y) on the parabola y² = 12x is given by:

For y² = 4ax, the focal distance of a point (x, y) is |x + a|. Here 4a = 12, so a = 3. The focal distance is x + 3 (since x ≥ 0 for this parabola).

Q5

The parabola x² = -24y opens:

Negative coefficient of y indicates downward opening.

Q6

For y² = 4ax, the length of latus rectum is:

The standard length of latus rectum is 4a.

Q7

Which condition ensures that the parabola y² = 4ax and the line x = c intersect at two distinct points?

Substituting x = c gives y² = 4ac. For y to have two distinct real roots, 4ac must be > 0. This means a and c must have the same sign, so ac > 0.

Q8

The focus of y² = 32x is:

4a = 32 gives a = 8. Focus = (8, 0).

Q9

A parabolic solar cooker is designed such that its focus is 8 cm from the vertex. If the width of the cooker at its open face is 40 cm, what is its depth?

Given a = 8. The equation is x² = 4ay → x² = 32y (assuming it opens upwards). Width is 40, so x = 20 at the rim. Substitute x = 20: 400 = 32y → y = 400/32 = 12.5 cm.

Q10

Which parabola opens towards the left?

Negative coefficient in y² = 4ax means opening left.

Q11

The directrix of y² = x is:

a = 1/4. Directrix is x = -1/4.

Q12

The endpoint of latus rectum of x² = 20y in the first quadrant is:

a = 5. Endpoints are (±10, 5).

Q13

A Diwali rocket follows a parabolic path given by y = 10x - x². What is the maximum height it achieves?

y = -(x² - 10x) = -(x - 5)² + 25. The maximum value of y occurs at x = 5, where y = 25. So the max height is 25 units.

Q14

The axis of symmetry of y² = 18x is:

The parabola y² = 4ax is symmetric about the x-axis.

Q15

The equation of a parabola with latus rectum length 16 opening right is:

Length = 4a = 16, so a = 4. Equation y² = 16x.