Identify the center of the ellipse with equation x^2+2y^2-10x+8y+25=0

(5, -2)

Step-by-step explanation:

The equation of the ellipse is given by:

where is the center of the ellipse.

The given equation of the ellipse is:

x² + 2y² - 10x + 8y + 25 = 0.

We have to reduce to the standard form so that we can compare and determine the center of the ellipse.

Subtracting from the equation on both sides, we get:

x² - 10x + 2y² + 8y + 25 - 25 = - 25

The next step would be complete the squares. Let us complete first.

Now for .

Therefore the equation becomes:

(x - 5)² + 2(y + 2)² = -25 + 25 + 8

⇒ (x - 5)² + 2(y - 2)² = 8

Since the needs 1, we divide the entire equation by 8.

⇒

Comparing with standard form we get:

and

⇒ (h,k) = (5,-2)

∴ The center of the ellipse is (5,-2).

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12