Advertisements
Advertisements
प्रश्न
If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.
उत्तर
Given that: Vertex = (0, 4) and Focus = (0, 2)
Let P(x, y) be any point on the parabola.
PB is perpendicular to the directrix.
We have PF = PB
⇒ `sqrt((x - 0)^2 + (y - 2)^2) = |(0 + y - 6)/sqrt(0 + 1)|`
⇒ `sqrt(x^2 + (y - 2)^2) = (y - 6)` .......[Equation of directrix is y = 6]
Squaring both sides, we have
x2 + (y – 2)2 = (y – 6)2
⇒ x2 + y2 + 4 – 4y = y2 + 36 – 12y
⇒ x2 – 4y + 12y – 32 = 0
⇒ x2 + 8y – 32 = 0
Hence, the required equation is x2 + 8y = 32.
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:
Focus (6, 0); directrix x = –6
Find the equation of the parabola that satisfies the following condition:
Vertex (0, 0) passing through (2, 3) and axis is along x-axis
Find the equation of the parabola that satisfies the following condition:
Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-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?
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 (0, 0) and the directrix 2x − y − 1 = 0
Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x − 4y + 3 = 0. Also, find the length of its latus-rectum.
Find the equation of the parabola if
the focus is at (0, −3) and the vertex is at (0, 0)
Find the equation of the parabola if the focus is at (a, 0) and the vertex is at (a', 0)
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 a parabola with vertex at the origin and the directrix, y = 2.
Find the equations of the lines joining the vertex of the parabola y2 = 6x to the point on it which have abscissa 24.
Find the coordinates of points on the parabola y2 = 8x whose focal distance is 4.
The line 2x − y + 4 = 0 cuts the parabola y2 = 8x in P and Q. The mid-point of PQ is
The equation 16x2 + y2 + 8xy − 74x − 78y + 212 = 0 represents
If the coordinates of the vertex and the focus of a parabola are (−1, 1) and (2, 3) respectively, then the equation of its directrix is
The locus of the points of trisection of the double ordinates of a parabola is a
The equation of the parabola whose focus is (1, −1) and the directrix is x + y + 7 = 0 is
Find the coordinates of a point on the parabola y2 = 8x whose focal distance is 4.
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 following parabolas:
Vertex at (0, 4), focus at (0, 2)
Find the equation of the following parabolas:
Focus at (–1, –2), directrix x – 2y + 3 = 0
Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.
The line lx + my + n = 0 will touch the parabola y2 = 4ax if ln = am2.
The equation of the parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is ______.
If the vertex of the parabola is the point (–3, 0) and the directrix is the line x + 5 = 0, then its equation is ______.
