Advertisements
Advertisements
Question
Answer the following:
Show that the two tangents drawn to the parabola y2 = 24x from the point (−6, 9) are at the right angle
Advertisements
Solution
Given equation of the parabola is y2 = 24x
Comparing this equation with y2 = 4ax, we get
4a = 24
∴ a = `24/4` = 6
Equation of tangent to the parabola y2 = 4ax having slope m is y = `"m"x + "a"/"m"`.
∴ y = `"m"x + 6/"m"`
But, (– 6, 9) lies on the tangent
∴ 9 = `-6"m" + 6/"m"`
∴ 9m = – 6m2 + 6
∴ 6m2 + 9m – 6 = 0
The roots m1 and m2 of this quadratic equation are the slopes of the tangents.
∴ m1m2 = `(-6)/6` = – 1
∴ Tangents drawn to the parabola y2 = 24x from the point (– 6, 9) are at right angle.
Alternate method:
Comparing the given equation with y2 = 4ax, we get
4a = 24
∴ a = 6
Equation of the directrix is x = – 6.
The given point lies on the directrix.
Since tangents are drawn from a point on the directrix are perpendicular,
Tangents drawn to the parabola y2 = 24x from the point (– 6, 9) are at the right angle.
APPEARS IN
RELATED QUESTIONS
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 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 whose vertex is O(0, 0) and focus at (–7, 0).
For the parabola 3y2 = 16x, find the parameter of the point (27, –12).
For the parabola y2 = 4x, find the coordinate of the point whose focal distance is 17
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)
If the tangent drawn from the point (–6, 9) to the parabola y2 = kx are perpendicular to each other, find k
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 common tangent to the parabola y2 = 4x and x2 = 32y
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 coordinates of a point on the parabola y2 = 8x whose focal distance is 4 are _______
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 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 2
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:
Find the equation of the tangent to the parabola y2 = 8x at t = 1 on it
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
x2 − y2 = 16
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 ______.
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 ______.
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))` ______.
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 equation to the line touching both the parabolas y2 = 4x and x2 = –32y 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 ______.
The equation of the line touching both the parabolas y2 = x and x2 = y 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 ______.
