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
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:
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 the equation of the parabola with vertex at the origin, axis along Y-axis and passing through the point (–10, –5).
For the parabola 3y2 = 16x, find the parameter of the point (3, – 4).
Find coordinates of the point on the parabola. Also, find focal distance.
y2 = 12x whose parameter is `1/3`
Find coordinates of the point on the parabola. Also, find focal distance.
2y2 = 7x whose parameter is –2
For the parabola y2 = 4x, find the coordinate of the point whose focal distance is 17
Find length of latus rectum of the parabola y2 = 4ax passing through the point (2, –6)
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 common tangent to the parabola y2 = 4x and x2 = 32y
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:
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 _______
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 = 8x at t = 1 on it
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 tangent at point P on the parabola y2 = 4ax meets the y-axis in Q. If S is the focus, show that SP subtends a right angle at Q
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 length of latus-rectum of the parabola x2 + 2y = 8x - 7 is ______.
The area of the triangle formed by the lines joining vertex of the parabola x2 = 12y to the extremities of its latus rectum 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 ______.
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 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))` ______.
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 of the line touching both the parabolas y2 = x and x2 = y 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 ______.
Area of the equilateral triangle inscribed in the circle x2 + y2 – 7x + 9y + 5 = 0 is ______.
The cartesian co-ordinates of the point on the parabola y2 = –16x, whose parameter is `1/2`, are ______.
