In parabola, give any example of a quadratic equation which doesnot touches the x-axis and y-axis

here is an example

(Good ol' Wolfram, one of my favourite webpages, never fails to amaze me)

I started with

x^2 + y^2 - 8x - 8y - 2xy = 0

http://www.wolframalpha.com/input/?i=plot+x%5E2+%2B+y%5E2+-+8x+-+8y+-+2xy+%3D+0+

As you can see it cuts both the x and the y axes
so I tweeted it and came up with

http://www.wolframalpha.com/input/?i=plot+%28x-3%29%5E2+%2B+%28y-3%29%5E2+-+4%28x-3%29+-+4%28y-3%29+-+2%28x-3%29%28y-3%29+%3D+0
using the equation:
(x-3)^2 + (y-3)^2 - 4(x-3) - 4(y-3) - 2(x-3)(y-3) = 0

which is just a translation of my first of
3 units to the right , and 3 units up

I also changed the coefficients of the x and y terms from 8 to 4, while toying with the shape and forgot to change them back, the concept is still there

To find an example of a quadratic equation that does not touch the x-axis and y-axis, we need to consider a parabola that opens upward or downward and is positioned in such a way that its vertex is not on either axis.

Let's consider the equation of a parabola that opens upward and is shifted upward by 2 units:

y = x^2 + 2

In this equation, the coefficient of x^2 is positive (+1), indicating an upward-opening parabola. Additionally, the constant term (2) shifts the parabola 2 units upward.

To verify that the parabola does not touch the x-axis, we can check whether it has any real roots. We can do this by setting y to 0 and solving for x:

0 = x^2 + 2

Rearranging the equation:

x^2 = -2

Since the discriminant (b^2 - 4ac) is negative (-2), the quadratic equation has no real roots. Therefore, the parabola defined by y = x^2 + 2 does not intersect the x-axis.

Similarly, to confirm that the parabola does not touch the y-axis, we can check whether it has any real x-intercepts. In this case, we can see that the vertex of the parabola is at (0, 2), indicating that it does not intersect the y-axis.

Therefore, the quadratic equation y = x^2 + 2 is an example of a parabola that does not touch the x-axis or y-axis.