Home
Class 12
MATHS
How many tangents to the circle x^2 + y...

How many tangents to the circle `x^2 + y^2 = 3` are normal to the ellipse `x^2/9+y^2/4=1?`

A

A. 3

B

B. 2

C

C. 1

D

D. 0

Text Solution

Verified by Experts

The correct Answer is:
D
Equation of normal to ellipse `(x^(2))/(9) + (y^(2))/(4) =1` at `P(3 cos theta, 2 sin theta)` is
`3x sec theta -2y cos theta =5`
If it is tangent to circle `x^(2) + y^(2) =3`, then
`rArr (5)/(sqrt(9sec^(2) theta+4 cosec^(2)theta)) =sqrt(3)`
`9 sec^(2) theta + 4 cosec^(2) theta = 9 = 4+ 9 tan^(2) theta + 4 cot^(2) theta = 25 +(3 tan theta -2 cot theta)^(2)`
`:. (9 sec^(2) theta + 4 cosec^(2) theta)_(min) =25`
`:.` no such `theta` exists.
Hence no tangents to circle which is normal to ellipse.
Promotional Banner

Topper's Solved these Questions

  • ELLIPSE

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|6 Videos

  • DOT PRODUCT

    CENGAGE PUBLICATION|Exercise DPP 2.1|15 Videos

  • EQAUTION OF STRAIGHT LINE AND ITS APPLICATION

    CENGAGE PUBLICATION|Exercise DPP 3.2|13 Videos

Similar Questions

Explore conceptually related problems

Find the equation of the common tangent in the first quadrant of the circle x^2+y^2=16 and the ellipse x^2/25+y^2/4=1 .Also find the length of the intercept of the tangent between the coordinates axes.

Tangents are drawn from any point on the circle x^(2)+y^(2) = 41 to the ellipse (x^(2))/(25)+(y^(2))/(16) =1 then the angle between the two tangents is

From an arbitrary point P on the circle x^2+y^2=9 , tangents are drawn to the circle x^2+y^2=1 , which meet x^2+y^2=9 at A and B . The locus of the point of intersection of tangents at A and B to the circle x^2+y^2=9 is (a) x^2+y^2=((27)/7)^2 (b) x^2-y^2((27)/7)^2 (c) y^2-x^2=((27)/7)^2 (d) none of these

If the chord of contact of tangents from a point on the circle x^(2) + y^(2) = a^(2) to the circle x^(2)+ y^(2)= b^(2) touches the circle x^(2) + y^(2) = c^(2) , then a, b, c are in-

A tangent at a point on the circle x^2+y^2=a^2 intersects a concentric circle C at two points Pa n dQ . The tangents to the circle X at Pa n dQ meet at a point on the circle x^2+y^2=b^2dot Then the equation of the circle is (a) x^2+y^2=a b (b) x^2+y^2=(a-b)^2 (c) x^2+y^2=(a+b)^2 (d) x^2+y^2=a^2+b^2

The circle (x-4)^(2) + (y-3)^(2) = 9 -

If the line a x+b y=2 is a normal to the circle x^2+y^2-4x-4y=0 and a tangent to the circle x^2+y^2=1 , then a and b are

A tangent is drawn to the ellipse to cut the ellipse x^2/a^2+y^2/b^2=1 and to cut the ellipse x^2/c^2+y^2/d^2=1 at the points P and Q. If the tangents are at right angles, then the value of (a^2/c^2)+(b^2/d^2) is

Find the common tangents to the hyperbola x^(2)-2y^(2)=4 and the circle x^(2)+y^(2)=1

Statement 1: An equation of a common tangent to the parabola y^2=16sqrt(3)x and the ellipse 2x^2+""y^2=""4 ""i s ""y""=""2x""+""2sqrt(3) . Statement 2: If the line y""=""m x""+(4sqrt(3))/m ,(m!=0) is a common tangent to the parabola y^2=""16sqrt(3)x and the ellipse 2x^2+""y^2=""4 , then m satisfies m^4+""2m^2=""24 .