Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
As shown in the figure APQ denotes the equilateral triangle with its equal sides of length l (say).
Here AP = l
So AR = l cos30°
= `l sqrt(3)/2`
Also, PR = `l sin 30^circ = l/2`.
Thus `(lsqrt(3))/2, l/2` are the coordinates of the point P lying on the parabola y2 = 4ax.
Therefore, `l^2/4 = 4a (lsqrt(3))/2`
⇒ `l = 8 asqrt(3)`.
THus, 8 `asqrt(3)` is the required length of the side of the equilateral triangle inscribed in the parabola y2 = 4ax.
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 coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum.
y2 = – 8x
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:
Focus (0, –3); directrix y = 3
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.
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 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 (−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)
At what point of the parabola x2 = 9y is the abscissa three times that of ordinate?
Find the equation of the parabola whose focus is (5, 2) and having vertex at (3, 2).
Find the equations of the lines joining the vertex of the parabola y2 = 6x to the point on it which have abscissa 24.
Write the equation of the parabola with focus (0, 0) and directrix x + y − 4 = 0.
Write the equation of the parabola whose vertex is at (−3,0) and the directrix is x + 5 = 0.
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 the point (2, 3) and directrix is the line x – 4y + 3 = 0 is ______.
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 following parabolas:
Directrix x = 0, focus at (6, 0)
Find the equation of the following parabolas:
Vertex at (0, 4), focus at (0, 2)
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 parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is ______.
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 ______.
