Solve a Parabola with Fraction

Parabola is a half circle and a curve formed by an intersection of right circular cone. A set of all points are same distance from a fixed line is called as directrix and also fixed point called as focus not in the directrix. The midpoint between the directrix and focus of the parabola is called as vertex and the line passing through the vertex and focus is called as axis. Let us see about how to get the value of vertex, latus rectum, focus and axis of symmetry in parabola equation. Let us see about solve a parabola with fraction in this article

The general form the parabolic curve is y = ax^(2) + bx + c or  y^(2) = 4ax . Substitute the above formula to find the vertices, latus rectum, focus and axis of symmetry.

Example 1 for Solve a Parabola with Fraction – Vertex:

Find the vertex of a parabola where y = x^ (2) + 7/2 x + 3

Solution:

Given parabola equation is y = x^ (2) + 7/2 x + 3

To find the vertices of a given parabola, we have to plug y = 0 in the above equation, we get,

0 = x^ (2) + 7/2 x + 3

Now we have to factor the above equation, we get,

So x^ (2) + 3/2x + 2x + 3 = 0

x(x + 3/2) + 2 (x + 3/2) = 0

(x + 3/2) (x + 2) = 0

From this x + 3/2 = 0 and x + 2 = 0

Then x = - 3/2 and x = - 2

So, the vertices of given parabola equation is (-3/2, 0) and (-2, 0) .

Example 2 for Solve a Parabola with Fraction – Focus:

What is the focus of the following parabola equation where y^ (2) = 10x

Solution:

Given parabola equation is y^ (2) = 10x is of the form y^ (2) = 4ax

To find the focus of a parabola use the formula to find the value of focus value.

We know that the formula for focus, p = 1 / (4a)

Now compare the given equation y^(2) = 10x with the general equation y^ (2) = 4ax . So, that 4a = 10

From this p = 1 / (4a) = 1 / 10

So, the focus of a parabola equation is (0, 1/10) .

Other Example Problems to Solve a Parabola Fraction

Example 3 for Solve a Parabola with Fraction – Axis of Symmetry:

What is the axis of symmetry of the parabola where y = 4x^ (2) + 8x + 14 ?
Solution:

Given parabolic curve equation is y = 4x^ (2) + 8x + 14

From the above equation, a = 4 and b = 8

So the axis of the symmetry of the given parabola is -b/ (2a) = -8/ (2 xx 4) = -8/8 = - 1

Therefore, the axis of symmetry for a given parabolic curve equation is -1 .

Example 4 for Solve a Parabola with Fraction – Latus Rectum:

Find the latus rectum of the given parabola equation y^ (2) = 6x

Solution:

The given parabola equation is y^ (2) = 6x

To find the latus rectum, we have to find the value of p.

The parabola equation is of the form y^ (2) = 4a

Here 4a = 6

So, p = 1/ (4a) = 1/6

The formula for latus rectum is 4p .

From this, the latus rectum of the parabola is = 4p = 4 (1/6) = 4/6 = 2/3

Therefore, the latus rectum for the parabola equation is 2/3 .