Advertisements
Advertisements
प्रश्न
If the line y = mx + 1 is tangent to the parabola y2 = 4x, then find the value of m.
उत्तर
We have y2 = 4x
Substituting the value of y = mx + 1 in y2 = 4x, we get
(mx + 1)2 = 4x
⇒ m2x2 + 2mx + 1 = 4x
⇒ m2x2 + (2m − 4)x + 1 = 0 .....(1)
Since, a tangent touches the curve at a point, the roots of (1) must be equal.
∴ D = 0
⇒ (2m − 4)2 − 4m2 = 0
⇒ 4m2 −16m + 16 − 4m2 = 0
⇒ m = 1
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 (0, –3); directrix y = 3
Find the equation of the parabola that satisfies the following condition:
Vertex (0, 0); focus (3, 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
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 (1, 1) and the directrix is x + y + 1 = 0
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, 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).
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 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 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 equation of the parabola whose focus is (1, −1) and the directrix is x + y + 7 = 0 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 ______.
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:
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 line lx + my + n = 0 will touch the parabola y2 = 4ax if ln = am2.
The equation of the parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is ______.
The equation of the ellipse whose focus is (1, –1), the directrix the line x – y – 3 = 0 and eccentricity `1/2` is ______.
