The Standard form of a quadratic equation is?

Consider the paraboloid z=x2+y2. The plane 8x−5y+z−2=0 cuts the paraboloid, its intersection being a curve. Find "the natural" p

Jet001 [13]

Answer:

The parametrization of the curve on the surface is

$c(t) = [x(t) , y(t), z(t)] \equiv [\frac{\sqrt{97} }{2} cost - 4 , \frac{\sqrt{97} }{2} sint + \frac{5}{2} , 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2} ]$

Where

$x = \frac{\sqrt{97} }{2} cost - 4$

$y = \frac{\sqrt{97} }{2} sint + \frac{5}{2}$

$z = 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2}$

Step-by-step explanation:

From the question we are told that

The equation for the paraboloid is $z = x^2 + y^2$

The equation of the plane is $8x - 5y + z -2 = 0$

Form the equation of the plane we have that

$z = 5y -8x +2$

So

$x^2 + y^2 = 5y -8x +2$

=> $x^2 + 8x + y^2 -5y = 2$

Using completing the square method to evaluate the quadratic equation we have

$(x + 4)^2 + (y - \frac{5}{2} )^2 = 2 +(\frac{5}{2} )^2 + 4^2$

$(x + 4)^2 + (y - \frac{5}{2} )^2 = \frac{97}{4}$

$(x + 4)^2 + (y - \frac{5}{2} )^2 = ( \frac{\sqrt{97} }{2} )^2$

representing the above equation in parametric form

$(x + 4) = \frac{\sqrt{97} }{2} cost$ , $(y -\frac{5}{2} ) = \frac{\sqrt{97} }{2} sin t$

$x = \frac{\sqrt{97} }{2} cost - 4$

$y = \frac{\sqrt{97} }{2} sint + \frac{5}{2}$

So from $z = 5y -8x +2$

$z = 5[\frac{\sqrt{97} }{2} sint + \frac{5}{2}] -8[ \frac{\sqrt{97} }{2} cost - 4] +2$

$z = 5\frac{\sqrt{97} }{2} sint + \frac{25}{2} -8 \frac{\sqrt{97} }{2} cost + 32 +2$

$z = 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2}$

Generally the parametrization of the curve on the surface is mathematically represented as

$c(t) = [x(t) , y(t), z(t)] \equiv [\frac{\sqrt{97} }{2} cost - 4 , \frac{\sqrt{97} }{2} sint + \frac{5}{2} , 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2} ]$

3 0

3 years ago

Please help me thanks sooo much

natima [27]

Answer:

P = 21x + 4

Step-by-step explanation:

The perimeter is the length of all the sides added up.

(3x) + (4x + 1) + (2x - 2) + (2x - 2) + (3x + 1) + (4x + 5) + (3x + 1)

3x + 4x + 1 + 2x - 2 + 2x - 2 + 3x + 1 + 4x + 5 + 3x + 1

21x + 1 - 2 - 2 + 1 + 5 + 1

21x + 4

The equation for the perimeter is

P = 21x + 4

6 0

3 years ago

A diver is at 20,m below sea level write his depth as a directed number<br><br>

Juliette [100K]

Answer:

-20

Step-by-step explanation:

20 Below -> -20

Think of it as a vertical number line, where sea level is 0. Anything above is positive, and anything below is negative.

7 0

3 years ago

A line intersects the points (6, -12) and (15, -3). What is the slope intercept equation for this line?

Korolek [52]

Let's say that the unknown value here is c. Although we are given two points on the line, it is only necessary to use one as there is only one unknown value, so let's say we choose the point (6, -12) and substitute this into the equation:

-12 = 1(6) - c
-12 - 6 = -c
-18 = -c
c = 18

Thus the equation for the line is y = 1x - 18

8 0

3 years ago

Which of the following is/are cube(s) of even number (i) 249 (ii) 50653

irinina [24]

Answer:

None (If there are just these two options i.e, 249 and 50653. I think there are more options, anyway)

Step-by-step explanation:

Cubes of even numbers are even and cube of odd numbers are odd.

Since both the numbers are odd, both of their cubes must be odd too.

Thank You

Hope this helps you

Please mark as brainliest

5 0

3 years ago