Advertisements
Advertisements
Question
If the line y = mx + 1 is tangent to the parabola y2 = 4x, then find the value of m.
Advertisements
Solution
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
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:
Vertex (0, 0) focus (–2, 0)
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 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 and the directrix, y = 2.
Find the equations of the lines joining the vertex of the parabola y2 = 6x to the point on it which have abscissa 24.
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.
The equation of the parabola whose vertex is (a, 0) and the directrix has the equation x + y = 3a, 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
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 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.
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 ______.
