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Question

Parabola: 4x^2 +24x+y= 58

Vertex:(__,___) (Exact) Focus:(____,____) (3decimals) Directrix: y=____(3decimals)

Length of Latus Rectum=_____(3decimals)

Answer 1

4x² + 24x + y = 58

It can be written as: (2x)² + 2(2x)(6) + 6² - 6² + y = 58

(2x + 6)² - 36 + y = 58

Add 36 to each side.

[2(x + 3)]²+ y = 94.

4(x+ 3)² + y = 94.

Subtract y from each side.

4(x+ 3)² = 94 - y

Multiply each side by negative one.

- 4(x+ 3)² = y - 94

4(-1)(x+ 3)² = (y - 94)

The parabola whose equation is y - k = 4p(x - h)²

opens upward if p is positive, and downward if p is negative. It has: vertex the point (h, k), focus the point (h, k+p), directrix, the horizontal line whose equation is y=k - p, and length of latus rectum = 4p,

4(-1)(x+ 3)² = (y - 94)

Here p = -1, h= - 3, k = 94

Vertex the point (h, k) = (-3, 94), focus the point (h, k+p) = (-3, 94-1) = (-3, 93)

Directrix, the horizontal line whose equation is y = k - p ⇒ y = 94 -(-1) = 95.

And length of latus rectum = 4p = 4(-1) = -4.

Comments

The parabola equation is 4x ² + 24x + y = 58.

Vertex of the parabola : (h, k) = (- 3, 94),

Directed distance from vertex to focus  = p =  - 1/16 = - 0.625,

Directrix : (y ) = k - p = 94 - (- 0.625) = 94.625,

Focus : (h , k + p ) = (- 3, 94 - 0.625 ) = (- 3, 93.937), and

Length of Latus Rectum = | 4p | units = | 4 * (- 1/16)| = 1/4 = 0.25 units.

Answer 2

Information About Parabolas :

Form of Equation	(x - h)2 = 4p (y - k)	(y - k)2 = 4p (x - h)
Vertex	(h, k )	(h, k )
Axis of Symmetry	x = h	y = k
Focus	(h, k + p )	(h + p, k )
Directrix	y = k - p	x = h - p
Direction of Opening	upward if a > 0, downward if a < 0	right if a > 0, left if a < 0
Length of Latus Rectum	| 4p | units	| 4p | units

The parabola equation is 4x ² + 24x + y = 58.

The standard form of parabola equation is (x - h)² = 4p (y - k), where (h, k) = vertex and p = directed distance from vertex to focus.

Write the equation : 4x ² + 24x + y = 58 in standard form of parabola.

4x ² + 24x = 58 - y

To change the expression [ 4x ² + 24x ] into a perfect square trinomial first divide the above equation by 4.

x ² + 6x = (58 - y)/4

To change the expression [ x ² + 6x ] into a perfect square trinomial add (half the x coefficient)² to each side of the expression

 Here x coefficient = 6. so, (half the x coefficient)² = (6/2)²= 9.

x ² + 6x + 9 = (58 - y )/4 + 9

(x + 3)² = - y/4 + 47/2

(x + 3)² = (- 1/4)(y - 94)

(x - (- 3))² = 4(- 1/16)(y - 94).

Compare the above equation with standard form of parabola.

Vertex : (h, k) = (- 3, 94),

Directed distance from vertex to focus  = p =  - 1/16 = - 0.625,

Directrix : (y ) = k - p = 94 - (- 0.625) = 94.625,

Focus : (h , k + p ) = (- 3, 94 - 0.625 ) = (- 3, 93.937), and

Length of Latus Rectum = | 4p | units = | 4 * (- 1/16)| = 1/4 = 0.25 units.