**Equation of parabola if vertex and focus is given :**

Here we are going to see how to find the equation of the parabola if vertex and focus is given.

**Step 1 :**

First we have to draw a rough diagram based on the given information. From this we come to know that the parabola is symmetric about which axis and it is open in which side.

**Step 2 :**

Distance between vertex and focus = a

**Step 3 :**

By applying these values in the standard form we will get the equation of the required parabola.

**Example 1 :**

Find the equation of the parabola if the vertex is (4, 1) and the focus is (4, − 3)

**Solution :**

From the given information the parabola is symmetric about y -axis and open downward

Distance between vertex and focus = a

VF = √(4-4)² + (1+3)²

= √0 + 4²

a = 4

vertex (h,k) ==> (4, 1)

(x - h)² = -4 a (y - k)

(x - 4)² = -4(4) (y - 1)

(x - 4)² = -16 (y - 1)

**Example 2 :**

Find the equation of the parabola if the vertex is (0, 0) and the focus is (0, − 4)

**Solution :**

From the given information the parabola is symmetric about y -axis and open downward

Distance between vertex and focus = a

VF = √(0-0)² + (0+4)²

= √0 + 4²

a = 4

vertex is (0, 0)

x² = -4 a y

x² = -4 (4) y

x² = -16 y

**Example 3 :**

Find the equation of the parabola if the vertex is (1, 4) and the focus is (-2, 4)

**Solution :**

From the given information the parabola is symmetric about x -axis and left downward

Distance between vertex and focus = a

VF = √(1+2)² + (4-4)²

= √3² + 0²

a = 3

vertex (h, k) ==> (1, 4)

(y-k)² = -4 a (x-h)

(y-4)² = -4 (3) (x-1)

(y-4)² = -12 (x-1)

After having gone through the stuff given above, we hope that the students would have understood "Equation of parabola if vertex and focus is given".

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