Advertisements
Advertisements
प्रश्न
Find the equation of tangent to the parabola y2 = 12x from the point (2, 5)
उत्तर
Given equation of the parabola is y2 = 12x.
Comparing this equation with y2 = 4ax, we get
4a = 12
∴ a = 3
Equation of tangent to the parabola y2 = 4ax having slope m is y = `"mx" + "a"/"m"`
Since the tangent passes through the point (2, 5),
5 = `2"m" + 3/"m"`
∴ 5m = 2m2 + 3
∴ 2m2 – 5m + 3 = 0
∴ 2m2 – 2m – 3m + 3 = 0
∴ 2m(m – 1) – 3(m – 1) = 0
∴ (m – 1)(2m – 3) = 0
∴ m = 1 or m = `3/2`
These are the slopes of the required tangents.
By slope point form, y – y1 = m(x – x1), the equations of the tangents are
y – 5 = 1(x – 2) and y – 5 = `3/2("x" - 2)`
∴ y – 5 = x – 2 and 2y – 10 = 3x – 6
∴ x – y + 3 = 0 and 3x – 2y + 4 = 0
APPEARS IN
संबंधित प्रश्न
Find co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:
x2 = –8y
Find the equation of the parabola with vertex at the origin, axis along Y-axis and passing through the point (–10, –5).
Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (3, 4)
For the parabola 3y2 = 16x, find the parameter of the point (3, – 4).
For the parabola 3y2 = 16x, find the parameter of the point (27, –12).
Find the focal distance of a point on the parabola y2 = 16x whose ordinate is 2 times the abscissa
Find the area of the triangle formed by the line joining the vertex of the parabola x2 = 12y to the end points of latus rectum.
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find its focus.
Two tangents to the parabola y2 = 8x meet the tangents at the vertex in the point P and Q. If PQ = 4, prove that the equation of the locus of the point of intersection of two tangent is y2 = 8(x + 2).
Find the equation of the locus of a point, the tangents from which to the parabola y2 = 18x are such that some of their slopes is –3
A circle whose centre is (4, –1) passes through the focus of the parabola x2 + 16y = 0.
Show that the circle touches the directrix of the parabola.
Select the correct option from the given alternatives:
If the focus of the parabola is (0, –3) its directrix is y = 3 then its equation is
Select the correct option from the given alternatives:
The coordinates of a point on the parabola y2 = 8x whose focal distance is 4 are _______
Select the correct option from the given alternatives:
If the parabola y2 = 4ax passes through (3, 2) then the length of its latus rectum is ________
Answer the following:
For the following parabola, find focus, equation of the directrix, length of the latus rectum, and ends of the latus rectum:
2y2 = 17x
Answer the following:
Find the equations of the tangents to the parabola y2 = 9x through the point (4, 10).
Answer the following:
Find the equation of the tangent to the parabola y2 = 8x which is parallel to the line 2x + 2y + 5 = 0. Find its point of contact
Answer the following:
A line touches the circle x2 + y2 = 2 and the parabola y2 = 8x. Show that its equation is y = ± (x + 2).
Answer the following:
Find the
(i) lengths of the principal axes
(ii) co-ordinates of the foci
(iii) equations of directrices
(iv) length of the latus rectum
(v) Distance between foci
(vi) distance between directrices of the curve
`x^2/144 - y^2/25` = 1
Answer the following:
Find the
(i) lengths of the principal axes
(ii) co-ordinates of the foci
(iii) equations of directrices
(iv) length of the latus rectum
(v) Distance between foci
(vi) distance between directrices of the curve
x2 − y2 = 16
The length of latus-rectum of the parabola x2 + 2y = 8x - 7 is ______.
The equation of the directrix of the parabola 3x2 = 16y is ________.
Let the tangent to the parabola S: y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then, the area (in sq.units) of the triangle PQR is equal to ______.
If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (–30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is ______.
If the line `y - sqrt(3)x + 3` = 0 cuts the parabola y2 = x + 2 at A and B, then PA. PB is equal to `("where coordinates of P are" (sqrt(3), 0))` ______.
The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) is ______.
Which of the following are not parametric coordinates of any point on the parabola y2 = 4ax?
If the vertex = (2, 0) and the extremities of the latus rectum are (3, 2) and (3, –2) then the equation of the parabola is ______.
A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola y = `(x - 1/4)^2 + α`, where α > 0. Then (4α – 8)2 is equal to ______.
Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is ______.
