Question

Define Parabola. What are the properties of Parabola? ​

Answer 1

Answer:

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Properties of Parabola. The parabola is symmetric about its axis. The axis is perpendicular to the directrix. The axis passes through the vertex and the focus. The tangent at vertex is parallel to the directrix.

Answer 2

Step-by-step explanation:

Def : parabola can be defined as it is the locus of a point which is equidistant from a fixed point S (focus) and a fixed line d(directrix of the parabola).

Derivations of y² = 4ax

y² = 4ax is the standard equation of parabola,

Derivations :

Let s be the focus and the line d be the directrix of the parabola. Draw a segment SZ perpendicular to the directrix d.

Take O, is the midpoint of seg ZS, From the origin, the X axis along the line OS and the line through O and perpendicular to the x axis as the y axis.

[ s and d are fixed, seg ZS is fixed ]

Let, ZS = 2a .

then S is (a , 0) and Z is (-a, 0)

therefore, the equation of the directrix d is x+a = 0 . Let P (x, y) be any point on the parabola, other than 0 . Draw a seg PM perpendicular to the directrix d.

Then from the focus directrix property of the parabola SP = PM -----(1)

SP =√[(x - a) ² + (y - 0)² and

PM = | x + a |

therefore,

√[ (x - a)² + y² ] = | x + a |

squaring on both the sides

x² - 2ax + a² + y² = x² + 2ax + a²

x² -x² + a² - a² + y² = 2ax + 2ax

y² = 4ax.