Question

Find the equation of a parabola through (-15/4 , 2)&(0,-1) and whose axis is parallel to the x-axis. and the latus ****** is equal 4.

Answer 1

Since it is a horizontal parabola.its equation has the for .

We have only two points.so there are two solutions.Say the points

The length of the latus rectum so a

Then the equation becomes

Plug the points into the equation.

 

Solve the system for b and c .

   ------------>(1)

       ------------->(2)

Add (1) and (2)

_________

Substitute the value in (2).

Plug the values into the equation

The equation of the parabola is .

Answer 2

Note :

• If the axis of symmetry is parallel to the x - axis, then it is a horizontal parabola.

 

• If the axis of symmetry is parallel to the y - axis, then it is a vertical parabola.
• The general form of a horizontal parabola is x = ay² + by + c.
• Latus rectum :

The line segment through the focus of a parabola and perpendicular to the axis of symmetry is called the latus rectum.

Length of the latus rectum = |1/a|.

• Since the axis is parallel to x - axis, it is a horizontal parabola.

Latus rectum = |1/a| = 4.

 1/a = ± 4

⇒ a = ± 1/4.

The points (- 15/4, 2) and (0, - 1) are passing through the parabola, means that, these are two solutions of the parabola.

Case 1 :

If a = 1/4, then the parabola equation is x = (1/4)y² + by + c.

Substitute the values of (x, y) = (- 15/4, 2) in the parabola equation.

- 15/4 = (1/4)(2)² + 2b + c

2b + c = - 15/4 - 1

8b + 4c = - 19  → ( 1 )

Substitute the values of (x, y) = (0, - 1) in the parabola equation.

0 = (1/4)(- 1)² + b(- 1) + c

- b + c = - 1/4

4b - 4c = 1    → ( 2 )

To find the values of b and c, solve eq (1) & eq( 2 ).

8b + 4c = - 19

4b - 4c = 1

( + )_____________

12b = -18

⇒ b = - 18/12 = - 3/2.

Substitute the value b = - 3/2 in eq (2), and solve for c.

4(- 3/2) - 4c = 1

4c = - 6 - 1

⇒ c = - 7/4.

Substitute the values b = - 3/2 and c = - 7/4 in parabola equation.

x = (1/4)y² + (- 3/2)y + (- 7/4).

The parabola equation is x = (1/4)y² - (3/2)y - (7/4).

Case 2 :

If a = - 1/4, then the parabola equation is x = (- 1/4)y² + by + c.

Substitute the values of (x, y) = (- 15/4, 2) in the parabola equation.

- 15/4 = (- 1/4)(2)² + 2b + c

2b + c = - 15/4 + 1

8b + 4c = - 11  → ( 1 )

Substitute the values of (x, y) = (0, - 1) in the parabola equation.

0 = (- 1/4)(- 1)² + b(- 1) + c

- b + c = 1/4

- 4b + 4c = 1

4b - 4c = - 1   → ( 2 )

To find the values of b and c, solve eq (1) & eq(2).

8b + 4c = - 11

4b - 4c = - 1

( + )_____________

12b = -12

⇒ b = - 12/12 = - 1.

Substitute the value b = - 1 in eq (2), and solve for c.

4(- 1) - 4c = - 1

4c = - 4 + 1

⇒ c = - 3.

Substitute the values b = - 1 and c = - 3 in parabola equation.

x = (- 1/4)y² + (- 1)y + (- 3).

The parabola equation is x = (- 1/4)y² - y - 3.