Advertisements
Advertisements
Question
The line lx + my + n = 0 will touch the parabola y2 = 4ax if ln = am2.
Options
True
False
Advertisements
Solution
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
RELATED QUESTIONS
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
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?
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 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 (0, 0)
Find the equation of the parabola if the focus is at (a, 0) and the vertex is at (a', 0)
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 line y = mx + 1 is tangent to the parabola y2 = 4x, then find the value of m.
PSQ is a focal chord of the parabola y2 = 8x. If SP = 6, then write SQ.
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 line 2x − y + 4 = 0 cuts the parabola y2 = 8x in P and Q. The mid-point of PQ is
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
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.
If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then find the equation of the parabola.
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:
Focus at (–1, –2), directrix x – 2y + 3 = 0
Find the equation of the set of all points the sum of whose distances from the points (3, 0) and (9, 0) is 12.
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 ______.
The equation of the ellipse whose focus is (1, –1), the directrix the line x – y – 3 = 0 and eccentricity `1/2` is ______.
