Advertisements
Advertisements
प्रश्न
Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.
उत्तर
Let (x, y) be any point.
Given points are (3, 0) and (9, 0)
We have `sqrt((x - 3)^2 + (y - 0)^2) + sqrt((x - 9)^2 + (y - 0)^2)` = 12
⇒ `sqrt(x^2 + 9 - 6x + y^2) + sqrt(x^2 + 81 - 18x + y^2)` = 12
Putting x2 + 9 – 6x + y2 = k
⇒ `sqrt(k) + sqrt(72 - 12x + k)` = 12
⇒ `sqrt(72 - 12x + k) = 12 - sqrt(k)`
Squaring both sides, we have
⇒ 72 – 12x + k = `144 + k - 24sqrt(k)`
⇒ `24sqrt(k)` = 144 – 72 + 12x
⇒ `24sqrt(k)` = 72 + 12x
⇒ `2sqrt(k)` = 6 + x
Again squaring both sides, we get
4k = 36 + x2 + 12x
Putting the value of k, we have
4(x2 + 9 – 6x + y2) = 36 + x2 + 12x
⇒ 4x2 + 36 – 24x + 4y2 = 36 + x2 + 12x
⇒ 3x2 + 4y2 – 36x = 0
Hence, the required equation is 3x2 + 4y2 – 36x = 0.
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) focus (–2, 0)
Find the equation of the parabola that satisfies the following condition:
Vertex (0, 0) passing through (2, 3) and axis is along x-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 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 (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 a parabola with vertex at the origin and the directrix, y = 2.
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.
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.
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
The locus of the points of trisection of the double ordinates of a parabola is a
The equation of the directrix of the parabola whose vertex and focus are (1, 4) and (2, 6) respectively 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 ______.
The equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0 is ______.
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 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 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 ______.
