Home
Class 12
MATHS
In the parabola y^2=4a x , then tangent ...

In the parabola `y^2=4a x ,` then tangent at `P` whose abscissa is equal to the latus rectum meets its axis at `T ,` and normal `P` cuts the curve again at `Qdot` Show that `P T: P Q=4: 5.`

Text Solution

Verified by Experts

Let P be `(at^(2),2at)`. Since `at^(2)=4a,t=pm2`. Consider t=2
and P(4a,4a). The tangent at P 2y=x+4a which meets the x-axis at T(-4a,0).
If the coordinates of Q are `(at_(1)^(2),2at_(1))`, then
`t_(1)=-t-(2)/(t)=-3`
So, Q is (9a,-6a). Therefore,
`(PQ)^(2)=125a^(2)and(PT)^(2)=80a^(2)`
`orPT:PQ=4:5`
Promotional Banner

Topper's Solved these Questions

  • PARABOLA

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.76|1 Videos

  • PARABOLA

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.77|1 Videos

  • PARABOLA

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.74|2 Videos

  • PAIR OF STRAIGHT LINES

    CENGAGE ENGLISH|Exercise Numberical Value Type|5 Videos

  • PERMUTATION AND COMBINATION

    CENGAGE ENGLISH|Exercise Comprehension|8 Videos

Similar Questions

Explore conceptually related problems

The two parabolas y^(2)=4x" and "x^(2)=4y intersect at a point P, whose abscissas is not zero, such that

The normal at P(ct,(c )/(t)) to the hyperbola xy=c^2 meets it again at P_1 . The normal at P_1 meets the curve at P_2 =

If the normals to the parabola y^2=4a x at the ends of the latus rectum meet the parabola at Q and Q^(prime), then Q Q^(prime) is (a)10 a (b) 4a (c) 20 a (d) 12 a

A tangent is drawn to the parabola y^2=4 x at the point P whose abscissa lies in the interval (1, 4). The maximum possible area of the triangle formed by the tangent at P , the ordinates of the point P , and the x-axis is equal to (a)8 (b) 16 (c) 24 (d) 32

If a normal to a parabola y^2 =4ax makes an angle phi with its axis, then it will cut the curve again at an angle

If a tangent to the parabola y^2=4a x meets the x-axis at T and intersects the tangents at vertex A at P , and rectangle T A P Q is completed, then find the locus of point Qdot

If a tangent to the parabola y^2=4a x meets the x-axis at T and intersects the tangents at vertex A at P , and rectangle T A P Q is completed, then find the locus of point Qdot

P is a point on the parabola whose ordinate equals its abscissa. A normal is drawn to the parabola at P to meet it again at Q. If S is the focus of the parabola, then the product of the slopes of SP and SQ is

Let N be the foot of perpendicular to the x-axis from point P on the parabola y^2=4a xdot A straight line is drawn parallel to the axis which bisects P N and cuts the curve at Q ; if N O meets the tangent at the vertex at a point then prove that A T=2/3P Ndot

If the tangent at a point P with parameter t , on the curve x=4t^2+3 , y=8t^3-1 t in R meets the curve again at a point Q, then the coordinates of Q are