Advertisements
Advertisements
Question
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.
Advertisements
Solution
Equation of parabola is y2 = 4ax
Let P(at2, 2at) be any point on the parabola.
In ΔPOA, we have
tan θ = `(2at)/(at^2) = 2/t`
⇒ t = `2/(tan theta)`
⇒ t = 2 cot θ .....(i)
OP = `sqrt((at^2 - 0)^2 + (2at - 0)^2)`
= `sqrt(a^2t^4 + 4a^2t^2)`
= `at sqrt(t^2 + 4)`
= `a xx 2 cot theta sqrt(4 cot^2 theta + 4)` .....[∵ t = 2 cot θ]
= `2a cot theta . 2sqrt(cot^2theta + 1)`
= 4a cot θ.cosec θ
= `4a * costheta/sintheta * 1/sintheta`
= `(4a costheta)/(sin^2theta)`
Hence, the required length = `(4a costheta)/(sin^2theta)`.
APPEARS IN
RELATED QUESTIONS
Find the equation of the parabola that satisfies the following condition:
Focus (0, –3); directrix y = 3
Find the equation of the parabola that satisfies the following condition:
Vertex (0, 0) focus (–2, 0)
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 (1, 1) and the directrix is x + 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 (a, 0) and the vertex is at (a', 0)
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).
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 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 parametric equations of a parabola are x = t2 + 1, y = 2t + 1. The cartesian equation of its directrix 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 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
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.
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.
If the line y = mx + 1 is tangent to the parabola y2 = 4x then find the value of m.
Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.
If the focus of a parabola is (0, –3) and its directrix is y = 3, then its equation 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 ______.
The equation of the ellipse whose focus is (1, –1), the directrix the line x – y – 3 = 0 and eccentricity `1/2` is ______.
