What conic section is represented by the polar equation r = 1 / 4 - 6cos theta

Option 2 - Hyperbola

Step-by-step explanation:

Given : The polar equation

To find : What conic section is represented by the polar equation?

Solution :

To find the conic section first we convert the polar into Cartesian equation    

We know,   and              

Substitute the value of r,

Squaring both side,

Applying completing the square we get,

This is in the form of hyperbola equation i.e.

Therefore, The given conic section is a hyperbola.

Hence, Option 2 is correct.

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for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.