Find the equation of parabola whose vertex is origin and directrix is x+y=2
Answers
Answer:
x² -2xy + y² -8x -8y = 0
Step-by-step explanation:
Given that x + y = 2 is the directrix of parabola,
Vertex is A(0, 0)
We know that perpendicular distance from vertex of the parabola to the
directrix is equal to 'a' where 4a is the Latus Rectum of the parabola.
Hence a = | 0 + 0 - 2|/√2 = √2
Thus, Latus Rectum = 4a = 4√2.
To find the equation of the parabola
We know that the line perpendicular to the directrix and passing through vertex is the equation of axis of parabola.
Hence, Any line perpendiculat ot x + y = 2 will be in the form x - y = k which passes through (0, 0) => k = 0
Hence x - y = 0 is the equation of the axis.
Equation of parabola in standard form is
(perpendicular distance from any point to axis of parabola)²
= latus rectum *(perpendicular distance from any point to perpendicular axis )
Line perpendicular to axis and passing though vertex(0, 0) will be x + y = 0,
hence equation of parabola will be
[(x - y)/√2]² = 4√2[(x + y )/√2]
⇒ (x - y)² = 8(x + y)
⇒ x² -2xy + y² -8x -8y = 0 is the required equation.
Hope, it helped !
Vertex is A(0, 0)
We know that perpendicular distance from vertex of the parabola to the
directrix is equal to 'a' where 4a is the Latus Rectum of the parabola.
Hence a = | 0 + 0 - 2|/√2 = √2
Thus, Latus Rectum = 4a = 4√2.
To find the equation of the parabola
We know that the line perpendicular to the directrix and passing through vertex is the equation of axis of parabola.
Hence, Any line perpendiculat ot x + y = 2 will be in the form x - y = k which passes through (0, 0) => k = 0
Hence x - y = 0 is the equation of the axis.
Equation of parabola in standard form is
(perpendicular distance from any point to axis of parabola)²
= latus rectum *(perpendicular distance from any point to perpendicular axis )
Line perpendicular to axis and passing though vertex(0, 0) will be x + y = 0,
hence equation of parabola will be
[(x - y)/√2]² = 4√2[(x + y )/√2]
⇒ (x - y)² = 8(x + y)
⇒ x² -2xy + y² -8x -8y = 0 is the required equation.
Hope, it helped !