Advertisements
Advertisements
प्रश्न
Answer the following:
A line touches the circle x2 + y2 = 2 and the parabola y2 = 8x. Show that its equation is y = ± (x + 2).
उत्तर
Given equation of the parabola is y2 = 8x
Comparing this equation with y2 = 4ax, we get
4a = 8
∴ a = 2
Equation of tangent to given parabola with slope m is y = `"m"x + 2/"m"`
∴ m2x – my + 2 = 0 ...(i)
Equation of the circle is x2 + y2 = 2
Its centre ≡ (0, 0) and Radius = `sqrt(2)`
Line (i) touches the circle.
∴ Length of perpendicular from the centre to the line (i) = radius
∴ `|("m"^2 0 - "m" 0 + 2)/sqrt("m"^4 + "m"^2)| = sqrt(2)`
∴ `4/("m"^4 + "m"^2)` = 2
∴ m4 + m2 – 2 = 0
∴ (m2 + 2)(m2 – 1) = 0
Since m2 ≠ – 2,
m2 – 1 = 0
∴ m = ± 1
When m = 1, equation of the tangent is
y = `(1)x + 2/((1))`
∴ y = (x + 2) ...(1)
When m = –1, equation of the tangent is
y = `(-1)x + 2/((-1))`
∴ y = –x – 2
∴ y = –(x + 2) ...(2)
From (1) and (2), the equation of the common tangents to the given parabola is y = ± (x + 2).
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 with vertex at the origin, axis along X-axis and passing through the point (2, 3)
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 length of latus rectum of the parabola y2 = 4ax passing through the point (2, –6)
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:
The length of latus rectum of the parabola x2 – 4x – 8y + 12 = 0 is _________
Select the correct option from the given alternatives:
Equation of the parabola with vertex at the origin and directrix x + 8 = 0 is __________
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 co-ordinates of a point of the parabola y2 = 8x having focal distance 10
Answer the following:
Find the equation of the tangent to the parabola y2 = 9x at the point (4, −6) on it
Answer the following:
Show that the two tangents drawn to the parabola y2 = 24x from the point (−6, 9) are at the right angle
Answer the following:
The slopes of the tangents drawn from P to the parabola y2 = 4ax are m1 and m2, show that m1 − m2 = k, where k is a constant.
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
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 P: y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of `π/4` with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is ______.
If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0)a ≠ 0, then' a' must be greater than ______.
If the normal at the point (1, 2) on the parabola y2 = 4x meets the parabola again at the point (t2, 2t), then t is equal to ______.
The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) 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 ______.
Let a variable point A be lying on the directrix of parabola y2 = 4ax (a > 0). Tangents AB and AC are drawn to the curve where B and C are points of contact of tangents. The locus of centroid of ΔABC is a conic whose length of latus rectum is λ, then `λ/"a"` 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 ______.
If vertex of a parabola is (2, –1) and the equation of its directrix is 4x – 3y = 21, then the length of its latus rectum is ______.
