Advertisements
Advertisements
प्रश्न
Answer the following:
The slopes of the tangents drawn from P to the parabola y2 = 4ax are m1 and m2, show that `("m"_1 /"m"_2)` = k, where k is a constant.
उत्तर
Let P(x1, y1) be any point on the parabola y2 = 4ax
Equation of tangent to the parabola y2 = 4ax having slope m is y = `"m"x + "a"/"m"`
This tangent passes through P(x1, y1)
∴ y1 = `"m"x_1 + "a"/"m"`
∴ my1 = m2x1 + a
∴ m2x1 – my1 + a = 0
This is a quadratic equation in ‘m’.
The roots m1 and m2 of this quadratic equation are the slopes of the tangents drawn from P.
∴ m1 + m2 = `y_1/x_1`, m1·m2 = `"a"/x_1`
Since (x1, y1) and a are constants, m1 m2 is a constant.
`("m"_1/"m"_2)` = k, where k is a constant.
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:
5y2 = 24x
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:
y2 = –20x
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:
3x2 = 8y
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 co-ordinate of focus, equation of directrix, length of latus rectum and the co-ordinate of end points of latus rectum of the parabola:
3y2 = –16x
Find the equation of the parabola whose vertex is O(0, 0) and focus at (–7, 0).
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 coordinates of the point on the parabola. Also, find focal distance.
y2 = 12x whose parameter is `1/3`
Find the equation of tangent to the parabola y2 = 12x from the point (2, 5)
Find the equation of common tangent to the parabola y2 = 4x and x2 = 32y
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
The tower of a bridge, hung in the form of a parabola have their tops 30 meters above the roadway and are 200 meters apart. If the cable is 5 meters above the roadway at the centre of the bridge, find the length of the vertical supporting cable 30 meters from the centre.
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 endpoints of latus rectum of the parabola y2 = 24x are _______
Select the correct option from the given alternatives:
The equation of the parabola having (2, 4) and (2, –4) as endpoints of its latus rectum is _________
Answer the following:
Find the Cartesian coordinates of the point on the parabola y2 = 12x whose parameter is 2
Answer the following:
Find the equation of the tangent to the parabola y2 = 9x at the point (4, −6) on it
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
16x2 + 25y2 = 400
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
The equation of the directrix of the parabola 3x2 = 16y is ________.
The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) is ______.
The equation to the line touching both the parabolas y2 = 4x and x2 = –32y is ______.
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 ______.
The equation of the parabola whose vertex and focus are on the positive side of the x-axis at distances a and b respectively from the origin 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 ______.
