Advertisements
Advertisements
प्रश्न
If the tangent drawn from the point (–6, 9) to the parabola y2 = kx are perpendicular to each other, find k
उत्तर
Given equation of the parabola is y2 = kx.
Comparing this equation with y2 = 4ax, we get
4a = k
∴ a = `"k"/4`
Equation of tangent to the parabola y2 = 4ax having slope m is y = `"m"x + "a"/"m"`
Since the tangent passes through the point (–6, 9),
9 = `- 6"m" + "k"/(4"m")`
∴ 36m = –24m2 + k
∴ 24m2 + 36m – k = 0
The roots m1 and m2 of this quadratic equation are the slopes of the tangents.
∴ m1m2 = `(-"k")/24`
Since the tangents are perpendicular to each other,
m1m2 = – 1
∴ `(-"k")/24` = – 1
∴ k = 24
Alternate method:
We know that, tangents drawn from a point on directrix are perpendicular.
∴ (–6, 9) lies on the directrix x = – a.
∴ – 6 = – a
∴ a = 6
Since 4a = k,
k = 4(6) = 24
APPEARS IN
संबंधित प्रश्न
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)
Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (1, –6)
Find the equation of the parabola with vertex at the origin, axis along X-axis and passing through the point (2, 3)
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.
Find coordinate of focus, vertex and equation of directrix and the axis of the parabola y = x2 – 2x + 3
Find the equation of tangent to the parabola y2 = 12x from the point (2, 5)
Find the equation of tangent to the parabola y2 = 36x from the point (2, 9)
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.
Select the correct option from the given alternatives:
The line y = mx + 1 is a tangent to the parabola y2 = 4x, if m is _______
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:
The coordinates of a point on the parabola y2 = 8x whose focal distance is 4 are _______
Select the correct option from the given alternatives:
The area of the triangle formed by the line joining the vertex of the parabola x2 = 12y to the endpoints of its latus rectum is _________
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:
Find the Cartesian coordinates of the point on the parabola y2 = 12x whose parameter is −3
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:
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:
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.
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 area of the triangle formed by the lines joining vertex of the parabola x2 = 12y to the extremities of its latus rectum is ______.
The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parabola, is another parabola whose directrix is ______.
Let y = mx + c, m > 0 be the focal chord of y2 = –64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of `4sqrt(2)` (m + c) 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 ______.
Through the vertex O of parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another, where P and Q are points on the parabola. If the locus of middle point of PQ is y2 = 2(x – l), then value of l is ______.
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 ______.
