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 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); 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.
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 (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 (−6, −6) and the vertex is at (−2, 2)
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).
PSQ is a focal chord of the parabola y2 = 8x. If SP = 6, then write SQ.
Write the equation of the parabola whose vertex is at (−3,0) and the directrix is x + 5 = 0.
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
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
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.
Find the coordinates of a point on the parabola y2 = 8x whose focal distance is 4.
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:
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 ______.
If the vertex of the parabola is the point (–3, 0) and the directrix is the line x + 5 = 0, then its equation is ______.
