Home
Class 12
MATHS
The locus of a point on the variable par...

The locus of a point on the variable parabola `y^2=4a x ,` whose distance from the focus is always equal to `k ,` is equal to (`a` is parameter)

Text Solution

Verified by Experts

The correct Answer is:
`4x^(2)+y^(2)-4kx=0`
Let the point be `(at^(2),2at)` and focus S be (a,0).
Now, `SP=at^(2)+a=k` (given) (1)
Let `(alpha,beta)` be the moving point. Then
`alpha=at^(2)andbeta=2at`
`rArr" "(alpha)/(beta)=(t)/(2)anda=(beta^(2))/(4alpha)" "(because "Point" (alpha,beta) "lies on" y^(2)=4ax)`
On substituting these value in equation (1), we get
`(beta^(2))/(4alpha)(1+(4alpha^(2))/(beta^(2)))=k`
`rArr" "beta^(2)+4alpha^(2)=4kalpha`
`rArr" "4x^(2)+y^(2)-4kx=0,`
which is the required locus.
Promotional Banner

Topper's Solved these Questions

  • PARABOLA

    CENGAGE ENGLISH|Exercise Concept Applications Exercise 5.2|17 Videos

  • PARABOLA

    CENGAGE ENGLISH|Exercise Concept Applications Exercise 5.3|7 Videos

  • PARABOLA

    CENGAGE ENGLISH|Exercise SOLVED EXAMPLES 5.14|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

Find the coordinates of a point the parabola y^(2)=8x whose distance from the focus is 10.

Find the coordinates of a point the parabola y^(2)=8x whose distance from the focus is 10.

The point on the parabola y^(2)=8x whose distance from the focus is 8 has x coordinate as

The coordinates of a point on the parabola y^2 =8x whose focal distance is 4, is

Find the coordinates of points on the parabola y^2=8x whose focal distance is 4.

Find the coordinates of points on the parabola y^2=8x whose focal distance is 4.

Find the coordinates of points on the parabola y^2=8x whose focal distance is 4.

The locus of the middle points of the chords of the parabola y^(2)=4ax which pass through the focus, is

Find the coordinates of the points on the parabola y^(2) = 4x whose focal distance is 5

The locus of the middle points of the focal chords of the parabola, y^2=4x is: