Home
Class 12
MATHS
A chord is drawn passing through P(2,2) ...

A chord is drawn passing through `P(2,2)` on the ellipse `(x^(2))/(25)+(y^(2))/(16) =1` such that it intersects the ellipse at A and B. Then maximum value of `PA.PB` is

A

`(61)/(4)`

B

`(59)/(4)`

C

`(71)/(4)`

D

`(63)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
B
Let equation of secant be
`(x-2)/(cos theta) = (y-2)/(sin theta) =r` (Parametric form)
Solving it with ellipse
`((r cos theta+2)^(2))/(25)+((r sin theta +2)^(2))/(16)=1`
`r^(2) (16 cos^(2) theta + 25 sin^(2) theta) +r (64 cos theta + 100 sin theta) - 236=0`
`|r_(1)r_(2)| = PA. PB = |(-236)/(16 cos^(2)theta+25 sin^(2)theta)|`
`=|-(236)/(16+9 sin^(2)theta)|`
Max. `PA. PB = (236)/(16) = (59)/(4)`
Promotional Banner

Topper's Solved these Questions

  • ELLIPSE

    CENGAGE|Exercise Multiple Correct Answers Type|6 Videos

  • DOT PRODUCT

    CENGAGE|Exercise DPP 2.1|15 Videos

  • ELLIPSE AND HYPERBOLA

    CENGAGE|Exercise Question Bank|1 Videos

Similar Questions

Explore conceptually related problems

The auxiliary circle of the ellipse x^(2)/25 + y^(2)/16 = 1

The eccentricity of ellipse (x^(2))/(25)+(y^(2))/(9)=1 is ……………. .

Two perpendicular tangents drawn to the ellipse (x^2)/(25)+(y^2)/(16)=1 intersect on the curve.

Tangents are drawn from the point P(3, 4) to the ellipse x^2/9+y^2/4=1 touching the ellipse at points A and B.

Find the maximum area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 which touches the line y=3x+2.

If from a point P(0, alpha ) , two normals other than the axes are drawn to the ellipse (x^2)/(25)+(y^2)/(16)=1 , such that | alpha | < k , then the value of 4k is______

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

AB and CD are two equal and parallel chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 . Tangents to the ellipse at A and B intersect at P and tangents at C and D at Q. The line PQ

The ellipse (x^(2))/(25)+(y^(2))/(16)=1 and the hyperbola (x^(2))/(25)-(y^(2))/(16) =1 have in common