Directrix MCQ Quiz in தமிழ் - Objective Question with Answer for Directrix - இலவச PDF ஐப் பதிவிறக்கவும்

Concept: 

Parabola,  y2 = 4ax, where a > 0, then  

Equation of directrix, x + a= 0

Parabola, y2 = - 4ax, where a > 0, then 

Equation of directrix, x - a= 0 

Calculation: 

Given parabolic equation, 3y2 = 16x 

⇒ y² = \(\frac{16}{3}\)x 

On comparing with standard equation,  y2 = 4ax , we get , a = \(\frac{4}{3}\) .

We know that equation of directrix, x + a= 0 

∴ Equation of directrix , x + \(\frac{4}{3}\) = 0 

⇒ 3x + 4 = 0 . 

The correct option is 5 .

Concept Used:

for \(y^2=4 a x\)

Directrix is x = -a, as directrix is the line for which each point on parabola is equidistance from directrix us well as focus.

Explanation

Point P(at²,2at) is lying on the curve y² = 6x.Therefore, point P(at2,2at) will satisfy the given curve.

⇒  (2at)²=6(at²)

⇒ 4a²t² =6at²

Therefore, a = 3/2

The equation of normal to the parabola y² = 4ax at point (x₁,y₁) is given by (y - y₁) = \(-\frac{y_1}{2a}\)(x-x₁)

Equation of normal at P(at2,2at) is

(y - 2at) = \(-\frac{2at}{2a}\)(x - at²)  

⇒ y = - tx + 2at + at³  ......(1)

Substitute the value of "a" in equation (1), and we get

⇒ y = - tx + 3t + \(\frac{3}{2}\)t3  

since normal at point P is passing through (5, -8), we get t = - 2

Coordinate of P : (6, - 6)

Since the slope of normal is  -2 therefore the slope of the tangent will be -1/2.

Therefore the equation of tangent passing through point P(6, - 6) and slope -1/2 is x + 2y + 6 = 0

Point Q (-3/2, y) is lying on the line x + 2y + 6 = 0

Therefore,  -3/2 + 2y + 6 = 0

⇒ y = -9/4 

CONCEPT:

The following are the properties of a parabola of the form: x2 = 4ay where a > 0

• Focus is given by (0, a)
• Vertex is given by (0, 0)
• Equation of directrix is given by: y = - a
• Equation of axis is given by: x = 0
• Length of latus rectum is given by: 4a
• Equation of latus rectum is given by: y = a

CALCULATION:

Given: Equation of parabola is x2 = 6y

The given equation of parabola can be re-written as: x2 = 4 ⋅ (3/2)y       --------(1)

Now by comparing the equation (1) with x2 = 4ay we get

⇒ a = 3/2

As we know that, the equation of directrix of the parabola of the form x2 = 4ay is given by: y = - a

So, the equation of directrix of the given parabola is: y = - 3/2

Hence, option C is the correct answer.

CONCEPT:

The following are the properties of a parabola of the form: x2 = - 4ay where a > 0

• Focus is given by (0, -a)
• Vertex is given by (0, 0)
• Equation of directrix is given by: y = a
• Equation of axis is given by: x = 0
• Length of latus rectum is given by: 4a
• Equation of latus rectum is given by: y = -a

CALCULATION:

Given: Equation of parabola is x2 = -16y

The given equation of parabola can be re-written as: x2 = -4 ⋅ 4y       ----(1)

Now by comparing the equation (1) with x2 = -4ay we get

⇒ a = 4

As we know that, equation of directrix of the parabola of the form x2 = - 4ay is given by: y = a

So, the equation of directrix of the given parabola is: y = 4

As we know that, equation of axis of the parabola of the form x2 = -4ay is given by: x = 0

Hence, option D is the correct answer.