FIND VERTEX FOCUS DIRECTRIX AND LATUS RECTUM OF PARABOLA

About "Find Vertex Focus Directrix and Latus Rectum of Parabola"

Find Vertex Focus Directrix and Latus Rectum of Parabola :

Here we are going to see some example problems to understand the concept of finding vertex, focus, directrix, equation of latus rectum of the parabola.

Find Vertex Focus Directrix and Latus Rectum of Parabola - Practice questions

Question 1 :

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

After having gone through the stuff given above, we hope that the students would have understood, "Find Vertex Focus Directrix and Latus Rectum of Parabola".

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