Advertisements
Advertisements
प्रश्न
Find the equation of the following parabolas:
Vertex at (0, 4), focus at (0, 2)
उत्तर
Given that vertex at (0, 4) and focus at (0, 2).
So, the equation of directrix is y – 6 = 0
According to the definition of the parabola
PF = PM.
`sqrt((x - 0)^2 + (y - 2)^2) = |y - 6|`
⇒ `sqrt(x^2 + y^2 + 4 - 4y) = |y - 6|`
Squaring both the sides, we get
x2 + y2 + 4 – 4y = y2 + 36 – 12y
⇒ x2 + 4 – 4y = 36 – 12y
⇒ x2 + 8y – 32 = 0
⇒ x2 = 32 – 8y
Hence, the required equation is x2 = 32 – 8y.
APPEARS IN
संबंधित प्रश्न
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum.
x2 = 6y
Find the equation of the parabola that satisfies the following condition:
Vertex (0, 0); focus (3, 0)
Find the equation of the parabola that satisfies the following condition:
Vertex (0, 0) passing through (2, 3) and axis is along x-axis
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
An equilateral triangle is inscribed in the parabola y2 = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
Find the equation of the parabola whose:
focus is (3, 0) and the directrix is 3x + 4y = 1
Find the equation of the parabola whose:
focus is (1, 1) and the directrix is x + y + 1 = 0
Find the equation of the parabola whose:
focus is (2, 3) and the directrix x − 4y + 3 = 0.
Find the equation of the parabola if
the focus is at (−6, −6) and the vertex is at (−2, 2)
Find the equation of the parabola if the focus is at (0, −3) and the vertex is at (−1, −3)
Find the equation of the parabola if the focus is at (0, 0) and vertex is at the intersection of the lines x + y = 1 and x − y = 3.
Find the equation of a parabola with vertex at the origin, the axis along x-axis and passing through (2, 3).
Find the equation of the parabola whose focus is (5, 2) and having vertex at (3, 2).
Find the coordinates of points on the parabola y2 = 8x whose focal distance is 4.
If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.
Write the equation of the directrix of the parabola x2 − 4x − 8y + 12 = 0.
PSQ is a focal chord of the parabola y2 = 8x. If SP = 6, then write SQ.
The parametric equations of a parabola are x = t2 + 1, y = 2t + 1. The cartesian equation of its directrix is
The line 2x − y + 4 = 0 cuts the parabola y2 = 8x in P and Q. The mid-point of PQ is
The equation of the directrix of the parabola whose vertex and focus are (1, 4) and (2, 6) respectively is
If V and S are respectively the vertex and focus of the parabola y2 + 6y + 2x + 5 = 0, then SV =
The equation of the parabola whose focus is (1, −1) and the directrix is x + y + 7 = 0 is
An equilateral triangle is inscribed in the parabola y2 = 4ax whose one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
Find the length of the line segment joining the vertex of the parabola y2 = 4ax and a point on the parabola where the line segment makes an angle θ to the x-axis.
Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.
Find the equation of the set of all points whose distance from (0, 4) are `2/3` of their distance from the line y = 9.
The equation of the ellipse whose focus is (1, –1), the directrix the line x – y – 3 = 0 and eccentricity `1/2` is ______.
