FIND VERTEX FOCUS DIRECTRIX AND LATUS RECTUM OF PARABOLA

Question :

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:

(i) y²  = 16x

Solution :

From the given information, we come to know that the given parabola is symmetric about x-axis and open right ward.

4a  =  16

a  =  4

Vertex : V (0, 0) 

Focus : F (a, 0)  ==>  F(4, 0)

Equation of latus rectum : x = a ==>  x  =  4

Equation of directrix : x = -a ==>  x  =  -4

Length of latus rectum  =  4a  =  4(4)  =  16 

Let us look into the next problems on "Find Vertex Focus Directrix and Latus Rectum of Parabola"

(ii) x² = 24y

Solution :

From the given information, we come to know that the given parabola is symmetric about y-axis and open upward.

4a  =  24

a  =  6

Vertex : V (0, 0) 

Focus : F (0, a)  ==>  F(0, 6)

Equation of latus rectum : y = a ==>  y  =  6

Equation of directrix : y = -a ==>  y  =  -6

Length of latus rectum  =  4a  =  4(6)  =  24

(iii)  y² = −8x

Solution :

From the given information, we know that the given parabola is symmetric about x-axis and left upward.

4a  =  8

a  =  2

Vertex : V (0, 0) 

Focus : F (-a, 0)  ==>  F(-2, 0)

Equation of latus rectum : x = -a ==>  x  =  -2

Equation of directrix : x = a ==>  x  =  2

Length of latus rectum  =  4a  =  4(2)  =  8

(iv) x² - 2x + 8y + 17  =  0

Solution :

x² - 2x + 8y + 17  =  0

x² - 2 ⋅ x  ⋅ 1 + 1² - 1² + 8y + 17  =  0

(x - 1)² - 1 + 8y + 17  =  0

(x - 1)²  =  -8y - 16

(x - 1)²  =  -8(y + 2)

The parabola is symmetric about y-axis and open downward.

4a  =  8

a  =  2

Vertex : V(h, k)  ==>  (1, -2)

Focus : F (h, k - a) ==>  F (1, -4)

Equation of directrix : y = k + a ==>  y  =  -2 + 2  =  0

Length of latus rectum  =  4a  =  4(2)  =  8

(v) y² - 4y - 8x + 12  =  0

Solution :

y² - 4y - 8x + 12  =  0

y² - 2 ⋅ y ⋅2 + 2² - 2² - 8x + 12  =  0

(y - 2)² - 4  =  8x - 12

(y - 2)²  =  8x - 8

(y - 2)²  =  8(x - 1)

The parabola is symmetric about x-axis and open rightward.

4a  =  8

a  =  2

Vertex : V(h, k)  ==>  (1, 2)

Focus : F (h+a, k) ==>  F (3, 2)

Equation of directrix : x = h - a ==>  x  =  -1

Length of latus rectum  =  4a  =  4(2)  =  8

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