NCERT Solutions: Exercise 10.2 - Conic Sections

Q1: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y² = 12x
Ans: The given equation is y² = 12x.
Here, the coefficient of x is positive. Hence, the parabola opens towards the right.
On comparing this equation with y² = 4ax, we obtain
4a = 12 ⇒ a = 3
∴ Coordinates of the focus = (a, 0) = (3, 0)
Since the given equation involves y², the axis of the parabola is the x-axis.
Equation of direcctrix, x = -a i.e., x = - 3 i.e., x + 3 = 0
Length of latus rectum = 4a = 4 × 3 = 12

Q2: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = 6y
Ans: The given equation is x² = 6y.
Here, the coefficient of y is positive. Hence, the parabola opens upwards.
On comparing this equation with x² = 4ay, we obtain

 
∴ Coordinates of the focus = (0, a) =
Since the given equation involves x², the axis of the parabola is the y-axis.
Equation of directrix,
Length of latus rectum = 4a = 6

Q3: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y² = - 8x
Ans: The given equation is y² = -8x.
Here, the coefficient of x is negative. Hence, the parabola opens towards the left.
On comparing this equation with y² = -4ax, we obtain
-4a = -8 ⇒ a = 2
∴ Coordinates of the focus = (-a, 0) = (-2, 0)
Since the given equation involves y², the axis of the parabola is the x-axis.
Equation of directrix, x = a i.e., x = 2
Length of latus rectum = 4a = 8

Q4: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = - 16y
Ans: The given equation is x² = -16y.
Here, the coefficient of y is negative. Hence, the parabola opens downwards.
On comparing this equation with x² = - 4ay, we obtain
-4a = -16 ⇒ a = 4
∴ Coordinates of the focus = (0, -a) = (0, -4)
Since the given equation involves x², the axis of the parabola is the y-axis.
Equation of directrix, y = a i.e., y = 4
Length of latus rectum = 4a = 16

Q5: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y² = 10x
Ans: The given equation is y² = 10x.
Here, the coefficient of x is positive. Hence, the parabola opens towards the right.
On comparing this equation with y² = 4ax, we obtain

 
∴ Coordinates of the focus = (a, 0)
Since the given equation involves y², the axis of the parabola is the x-axis.
Equation of directrix,
Length of latus rectum = 4a = 10

Q6: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for x² = -9y
Ans: The given equation is x² = -9y.
Here, the coefficient of y is negative. Hence, the parabola opens downwards.
On comparing this equation with x² = -4ay, we obtain

 
∴ Coordinates of the focus =
Since the given equation involves x², the axis of the parabola is the y-axis.
Equation of directrix,
Length of latus rectum = 4a = 9

Q7: Find the equation of the parabola that satisfies the following conditions: Focus (6, 0); directrix x = -6
Ans: Focus (6, 0); directrix, x = -6
Since the focus lies on the x-axis, the x-axis is the axis of the parabola.
Therefore, the equation of the parabola is either of the form y² = 4ax or
y² = - 4ax.
It is also seen that the directrix, x = -6 is to the left of the y-axis, while the focus (6, 0) is to the right of the y-axis. Hence, the parabola is of the form y² = 4ax.
Here, a = 6
Thus, the equation of the parabola is y² = 24x.

Q8: Find the equation of the parabola that satisfies the following conditions: Focus (0, -3); directrix y = 3
Ans: Focus = (0, -3); directrix y = 3
Since the focus lies on the y-axis, the y-axis is the axis of the parabola.
Therefore, the equation of the parabola is either of the form x² = 4ay or
x² = - 4ay.
It is also seen that the directrix, y = 3 is above the x-axis, while the focus
(0, -3) is below the x-axis. Hence, the parabola is of the form x² = -4ay.
Here, a = 3
Thus, the equation of the parabola is x² = -12y.

Q9: Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0); focus (3, 0)
Ans: Vertex (0, 0); focus (3, 0)
Since the vertex of the parabola is (0, 0) and the focus lies on the positive x-axis, x-axis is the axis of the parabola, while the equation of the parabola is of the form y² = 4ax.
Since the focus is (3, 0), a = 3.
Thus, the equation of the parabola is y² = 4 × 3 × x, i.e., y² = 12x

Q10: Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) focus (-2, 0)
Ans: Vertex (0, 0) focus (-2, 0)
Since the vertex of the parabola is (0, 0) and the focus lies on the negative x-axis, x-axis is the axis of the parabola, while the equation of the parabola is of the form y² = -4ax.
Since the focus is (-2, 0), a = 2.
Thus, the equation of the parabola is y² = -4(2)x, i.e., y² = -8x

Q11: Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) passing through (2, 3) and axis is along x-axis.
Ans: Since the vertex is (0, 0) and the axis of the parabola is the x-axis, the equation of the parabola is either of the form y² = 4ax or y² = -4ax.
The parabola passes through point (2, 3), which lies in the first quadrant.
Therefore, the equation of the parabola is of the form y² = 4ax, while point
(2, 3) must satisfy the equation y² = 4ax.
 
Thus, the equation of the parabola is

 

Q12: Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis
Ans: Since the vertex is (0, 0) and the parabola is symmetric about the y-axis, the equation of the parabola is either of the form x² = 4ay or x² = -4ay.
The parabola passes through point (5, 2), which lies in the first quadrant.
Therefore, the equation of the parabola is of the form x² = 4ay, while point
(5, 2) must satisfy the equation x² = 4ay.

 
Thus, the equation of the parabola is