Advertisements
Advertisements
प्रश्न
The line lx + my + n = 0 will touch the parabola y2 = 4ax if ln = am2.
पर्याय
True
False
Advertisements
उत्तर
This statement is True.
Explanation:
Given equation of parabola is y2 = 4ax .....(i)
And the equation of line is lx + my + n = 0 ......(ii)
From equation (ii), we have
y = `(-lx - n)/m`
Putting the value of y in equation (i) we get
`((-lx - n)/m)^2 = 4ax`
⇒ `l^2x^2 + n^2 + 2lnx - 4am^2x` = 0
⇒ `l^2x^2 + (2ln - 4am^2)x + n^2` = 0
If the line is the tangent to the circle
Then `b^2 - 4ac` = 0
`(2ln - 4am^2) - 4l^2n^2` = 0
⇒ `4l^2n^2 + 16a^2m^4 - 16lnm^2a - 4l^2n^2` = 0
⇒ `16a^2m^4 - 16lnm^2a` = 0
⇒ `16am^2(am^2 - ln)` = 0
⇒ `am^2(am^2 - ln)` = 0
⇒ `am^2 ≠ 0`
∴ `am^2 - ln` = 0
∴ `ln - am^2`
APPEARS IN
संबंधित प्रश्न
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:
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.
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
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 (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 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 (0, −3) and the vertex is at (0, 0)
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.
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, 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).
If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.
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 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 directrix of the parabola whose vertex and focus are (1, 4) and (2, 6) respectively is
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 ______.
The equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0 is ______.
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.
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 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.
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 ______.
