1. 2. 3. 4. 5. Find the vertex, the focus, and the directrix. Then draw the grap

Question

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Find the vertex, the focus, and the directrix. Then draw the graph. X = 2y2 The vertex of x = 2y2 is (Type an ordered pair.) The focus of x = 2y2 is (Type an ordered pair. Type an integer or a fraction.) The directrix is x = Choose the correct graph of x = 2y2. The parametric equation and parameter intervals for the motion of a particle in the xy - plane are given below. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. x = 3 cos (t), y = 3 sin (t), 0 t pi Choose the correct Cartesian equation that represents the same path as the parametric equations. x2 + y2 = 18 x2 + y2 = 9 (x + y)2= 9 (x + y )2= 18 Choose the correct graph that represents this motion. Find the area of the shaded region in the accompanying figure. Is the graph of r = 6 tan theta, - pi/2 theta pi/2, asymptotic to the lines x = 6 and x = - 6? The area of the shaded region in the accompanying figure is . (type an exact answer, using pi as needed.) Is the graph of r = 6 tan theta, - pi/2 theta pi/2, asymptotic to the lines x = 6 and x = - 6? yes no Identify the symmetries of the curve r = 8 + 7 sin theta. Then sketch the curve. Is the curve symmetric about the x-axis? No Yes Is the curve symmetric about the y-axis? Yes No Is the curve symmetric about the origin? Yes No Identify the sketch that shows the curve r = 8 + 7 sin theta. Find the eccentricity of the ellipse. Then find the ellipse's foci and directrices. 10x2 + 6y2 = 60 The eccentricity of the ellipse is (Type an exact answer, using radicals as needed.) The ellipse's foci are (Type ordered pairs. Use a comma to separate answers as needed.) Choose the correct equations of the directrices. x = plusminus 5 y = plusminus 6 0 y= plusminus 5 y = plusminus 6

Explanation / Answer

1) The given parabola is of the form y^2 = 4ax

Comparing with the standard form, y^2 = x/2

4a = 1/2

a = 1/8

The vertex of the parabola = (0,0)

Focus F = (a,0) = (1/8 , 0)

Directrix = (-a,0) = (-1/8, 0)

For the shape of the parabola, it is difficult to get a difference between C and D. However, I would guess the answer to be the broader parabola, i.e Option D as point (8,2) and (8,-2) are lying on it.

2) x = 3 cos (t), y = 3 sin (t)

Squaring and adding shall give us the required equation of the curve as it would eliminate the parameter t.

We get,

x^2 + y^2 = 9

Curve C gives the best representation since for the given range of parameter 't', x takes both positive and negative values, but y takes only positive values.

Hence Option C.

3) I doubt there is a mistake in problem 3. The graph of cosec (x) lies in the range

1 < cosec (x) < inf. However, the graph given touches the point 0 which is not possible. I guess you clarify the question and ask to me in the comments.

4) Te given curve is an equation of cardioid.

r = 8 + 7 sin (t)

The Curve is symmetric about x-axis

The curve is neither symmetric about y-axis nor about origin

Hence, Option D

5) the equation of ellipse in a standard form is:

x^2/ a^2 + y^2 / b^2 = 1

So, bringing it in a standard form,

x^2 /6 + y^2 / 10 = 1

eccentricity = sqrt ( 1 - a^2/b^2 ) = sqrt ( 1- 6/10) = 2 / sqrt (10)

Here, y-axis is the semi-major axis and x- axis is the minor axis.

So, foci = (0, be) and (0,-be)

foci = (0,2) and (0,-2)

Directrices => y = b/e = sqrt (10)/ [2/sqrt(10)] = 10/2 = 5

Directrices = y= 5 and y = -5.

Hope this answer helps. Please Rate it ASAP. It took me very long to work out.

Please reward and rate it carefully.

Thanks.   

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