Question

the focus of the parabola Y squared minus 4 Y - 8 x + 4 = 0 is​

Answer 1

Given :

Equation of parabola :

${y}^{2} - 4y - 8x + 4 = 0$

To find :

Focus of the parabola.

Solution :

Equation of the given parabola is :

$\bf {y}^{2} - 4y - 8x + 4 = 0$

(equation 1)

we have to put this equation in the form as:

$\bf Y^2 = 4aX \:$

(equation 2)

It is the general form of parabola.

so

$\bf {y}^{2} - 4y - 8x + 4 = 0 \\ {y}^{2} - 4y = 8x -4$

Adding +4 on both sides,

${y}^{2} - 4y + 4 = 8x - 4 + 4$

$\bf \: {(y - 2)}^{2} = 8x = 4 \times (2) \times x$

(equation 3)

On comparing equation 3 with general equation of parabola (equation 3)

we get

Y = y - 2

X = x

a = 2

We know that to get the focus of parabola.

At X = a, the value of Y should be 0.

So putting the value of X and Y in equation 3.

X = x = a = 2

$\bf \: {(y - 2)}^{2} = 4 \times (2) \times 2 = 16$

It means

y - 2 = ±4

so

y -2 = 4 and y - 2 = -4

so

y = 6 and y = -2

so focus of the given parabola are :

(2,6) and (2,-2).