Question

Define a parabola and derive the equation of the parabola y2 = 4ax​

Answer 1

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.