Advertisements
Advertisements
प्रश्न
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?
उत्तर
Its form is of the shape of a parabola.
Let OX, OY be its coordinate axis, and the equation is y2 = 4ax.
Height of arch, OL = 10 m
Width EF = 5 m
LF = `1/2` EF = `1/2 xx 5 = 5/2`
Coordinates of point F `(10, 5/2)`
Since the point `(10, 5/2)` lies on the parabola y2 = 4ax
∴ `(5/2)^2 = 4a xx 10` or `40a = 25/4`
∴ 4a = `25/4 xx 1/10 = 5/8`
∴ Equation of parabola y2 = `5/8 x`
2 m below top O, let the width of the arch be 2b.
∴ PM = `1/2 "PQ" = 1/2 xx 2"b" = "b"`
P has coordinates of the point (2, b) which lies on the parabola `"y"^2 = 5/8 "x"`.
∴ `"b"^2 = 5/8 xx 2 = 5/4`
∴ b = `sqrt5/2`
The width of the arch at this location,
= `2"b"`
= `2 xx sqrt5/2`
= `sqrt5` meter
= 2.24 meters (approximately)
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) 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 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 (0, 0) and the directrix 2x − 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, 0) and vertex is at the intersection of the lines x + y = 1 and x − y = 3.
At what point of the parabola x2 = 9y is the abscissa three times that of ordinate?
Find the equation of a parabola with vertex at the origin and the directrix, y = 2.
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest wire being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
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.
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.
Write the equation of the parabola with focus (0, 0) and directrix x + y − 4 = 0.
PSQ is a focal chord of the parabola y2 = 8x. If SP = 6, then write SQ.
Write the equation of the parabola whose vertex is at (−3,0) and the directrix is x + 5 = 0.
The equation of the parabola whose vertex is (a, 0) and the directrix has the equation x + y = 3a, is
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
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 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
The equations of the lines joining the vertex of the parabola y2 = 6x to the points on it which have abscissa 24 are ______.
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.
If the line y = mx + 1 is tangent to the parabola y2 = 4x then find the value of m.
Find the equation of the following parabolas:
Vertex at (0, 4), focus at (0, 2)
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.
If the vertex of the parabola is the point (–3, 0) and the directrix is the line x + 5 = 0, then its equation is ______.
