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.
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 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 (5, 2) and symmetric with respect to y-axis.
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
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 being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
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 (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.
Find the equations of the lines joining the vertex of the parabola y2 = 6x to the point on it which have abscissa 24.
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 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 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 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.
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 line lx + my + n = 0 will touch the parabola y2 = 4ax if ln = am2.
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 ______.
