Home
Class 12
MATHS
A circle with radius |a| and center on t...

A circle with radius `|a|` and center on the y-axis slied along it and a variable line through (a, 0) cuts the circle at points `Pa n dQ` . The region in which the point of intersection of the tangents to the circle at points `P` and `Q` lies is represented by (a) `y^2geq4(a x-a^2)` (b) `y^2lt=4(a x-a^2)` (c) `ygeq4(a x-a^2)` (d) `ylt=4(a x-a^2)`

A

`y^(2) ge 4a (x-a)`

B

`y^(2) le 4ax`

C

`x^(2) +y^(2) le 4a^(2)`

D

`x^(2) -y^(2) ge a^(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let the circle be `x^(2) + (y-alpha)^(2) =a^(2)`. Let the point of intersection of tangents at P and Q be (h,k).
Then equation of PQ, is `hx + (k-alpha) (y-alpha) -a^(2) =0`.
As PQ passes through (a,0), we have
`ha -alpha (k-alpha) -a^(2) =0`
`rArr alpha^(2) -k alpha +a (h-a) =0`
Now for real values of `alpha`, we have `D ge 0`.
`rArr k^(2) -4a(h-a) ge 0`
or `y^(2) ge 4a (x-a)`
Promotional Banner

Topper's Solved these Questions

  • CIRCLES

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|9 Videos

  • CIRCLES

    CENGAGE PUBLICATION|Exercise Comprehension Type|8 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

  • COMPLEX NUMBERS

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

Similar Questions

Explore conceptually related problems

If the tangent to the ellipse x^2+2y^2=1 at point P(1/(sqrt(2)),1/2) meets the auxiliary circle at point R and Q , then find the points of intersection of tangents to the circle at Q and Rdot

Find the point(s) of intersection of the line 2x + 3y = 18 and the circle x^2 + y^2 = 25

Find the point of intersection of the circle x^2+y^2-3x-4y+2=0 with the x-axis.

If the straight line x - 2y + 1 = 0 intersects the circle x^2 + y^2 = 25 at points P and Q, then find the coordinates of the point of intersection of the tangents drawn at P and Q to the circle x^2 + y^2 = 25 .

Tangents are drawn to the circle x^2+y^2=9 at the points where it is met by the circle x^2+y^2+3x+4y+2=0 . Find the point of intersection of these tangents.

If the tangents are drawn to the circle x^2+y^2=12 at the point where it meets the circle x^2+y^2-5x+3y-2=0, then find the point of intersection of these tangents.

Find the equation of the smallest circle passing through the point of intersection of the line x+y=1 and the circle x^(2)+y^(2)=9 .

If line x-2y-1=0 intersects parabola y^(2)=4x at P and Q, then find the point of intersection of normals at P and Q.

The line 2x-y+1=0 is tangent to the circle at the point (2, 5) and the center of the circle lies on x-2y=4 . Then find the radius of the circle.

The straight line y=mx+c cuts the circle x^2+y^2=a^2 at real points if