Home
Class 12
MATHS
IF three distinct normals to the parabol...

IF three distinct normals to the parabola `y^(2)-2y=4x-9` meet at point (h,k), then prove that `hgt4`.

Text Solution

Verified by Experts

Given parabola is `(y-1)^(2)=4(x-2)`.
Equation of normal to parabola having slope m is
`y-1=m(x-2)-2m-m^(3)`
It passes through (h,k)
`:." "k-1=m(h-2)-2m-m^(3)`
`or" "m^(3)+(4-h)m+k-1=0`
This equation has three distinct real roots.
`:." "3m^(2)+(4-h)=0` has two distinct real roots.
`:." "4-hlt0orhgt4`
Promotional Banner

Topper's Solved these Questions

  • PARABOLA

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.75|1 Videos

  • PARABOLA

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.76|1 Videos

  • PARABOLA

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.73|1 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

If three distinct normals can be drawn to the parabola y^(2)-2y=4x-9 from the point (2a, 0) then range of values of a is

If three distinct normals can be drawn to the parabola y^2-2y=4x-9 from the point (2a ,b) , then find the range of the value of adot

If normals drawn at three different point on the parabola y^(2)=4ax pass through the point (h,k), then show that h hgt2a .

The equation to the normal to the parabola y^(2)=4x at (1,2) is

The normal to parabola y^(2) =4ax from the point (5a, -2a) are

The normal to the parabola y^(2)=8x at the point (2, 4) meets the parabola again at eh point

The normal to the parabola y^(2)=8ax at the point (2, 4) meets the parabola again at the point

If the normals at P, Q, R of the parabola y^2=4ax meet in O and S be its focus, then prove that .SP . SQ . SR = a . (SO)^2 .

Find the number of distinct normals drawn to parabola y^(2) = 4x from the point (8, 4, sqrt(2))

The normals at P, Q, R on the parabola y^2 = 4ax meet in a point on the line y = c. Prove that the sides of the triangle PQR touch the parabola x^2 = 2cy.