Consider the equation x2 + y2 = 25, which is the equation of a circle. Explain what you might do in order to change this equation into the equation of each of the following: an ellipse
a hyperbola
a parabola
What real-life objects or physical models of an ellipse, hyperbola, and parabola?

1) You divide x measure by a2 and y measure by b^2 to equal 1 whole.

2) Using the vertices 0,0 we look for the negative (x-h^2)/ a2   and (y-k)^2/b^2

Which means distance from center to a vertices is 'a ' and means (hk)

3) So the parabola is a conic section (a section of a cone). Equations. x-squared is a parabola. The simplest equation for a parabola is y = x2.   Most renowned is the given y = ax2 + bx + c

For graphs the difference from the given x2 equation is we keep double 'x' x2 to y and multiply h to 4p;  we see (x - h)2 = 4p (y - k) usually focusing upright g k and p. Note below this focus changes from y to x if rotated on transversal.

Real life objects ;

Ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves.

Hyperbola;

Satellites, lenses radios, monitors Graphing a hyperbola shows this immediately: when the x-value is small, the y-value is large, and vice versa. Many real-life situations can be described by the hyperbola, including the relationship between the pressure and volume of a gas.

Parabolas; Are also used in satellite dishes to help reflect signals that then go to a receiver.From the paths of thrown baseballs, to fountains, this geometric shape is prevalent, and even functions to help focus light and radio waves.

Using Parabolic Reflectors to Focus Light

Flat surfaces scattered light too much to be useful to mariners. Spherical reflectors increased brightness, but could not give a powerful beam. But using a parabola-shaped reflector helped focus light into a beam that could be seen for long distances.

The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope.

The name "parabola" is due to Apollonius, who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve.

Where the focus–directrix property of the parabola and other conic sections is due to Pappus of Alexandria. (c. 340) an encyclopedist where 8 volumes survived.

Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

Step-by-step explanation:

The standard equation for an ellipse, x 2 / a 2 + y2 / b 2 = 1, represents an ellipse centered at the origin and with axes lying along the coordinate axes. In general, an ellipse may be centered at any point, or have axes not parallel to the coordinate axes.

The standard for a parabola is (x - h)2 = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p. If the parabola is rotated so that its vertex is (h,k) and its axis of symmetry is parallel to the x-axis, it has an equation of (y - k)2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p.

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step-by-step explanation: