Three points $A,B,C$ are taken on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with eccentric angles ${\rm{\theta }},{\rm{\theta }} + {\rm{\alpha }}\,{\rm{and }}\,{\rm{\theta }} + 2{\rm{\alpha }}$, then

The area $\Delta ABC$ is independent of $\theta $

The area $\Delta ABC$ is independent of $\alpha $

The maximum value of area is $\frac{{\sqrt 3 }}{4}ab$

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JEE MATHEMATICS MCQ: ELLIPSE

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Three points

A,B,C

are taken on the ellipse

\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1

with eccentric angles

\mathrm{\theta },\mathrm{\theta } + \mathrm{\alpha }\,\mathrm{and }\,\mathrm{\theta } + 2\mathrm{\alpha }

, then

View answer and explanation

Correct option: D

The maximum value of area is

\frac{{3\sqrt 3 }}{4}ab

Option D (The maximum value of area is

\frac{{3\sqrt 3 }}{4}ab

) is correct because the area of the triangle formed by the points on the ellipse is maximized when

\cos \alpha = -\frac{1}{2}

, resulting in the maximum area of

\frac{{3\sqrt 3 }}{4}ab

More Questions from ELLIPSE

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Let

\frac{x^2}{144}+\frac{y^2}{81}=1

be the equation of an ellipse. If

m

be the minimum value of the segment of tangent to the ellipse intercepted by the coordinate axes and

\mathrm{M}

be the maximum distance of the centre of the ellipse from the normal, then the value of

\mathrm{mM}

is equal to Let

P

be a variable point on the ellipse

\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1

with foci at

S

and

S'.

A

be the area of triangle

PSS',

then the maximum value of

A

is If the co-ordinates of two points

A

and

B

are

(\sqrt{7}, 0)

and

(-\sqrt{7}, 0)

respectively and

P

is any point on the conic,

9 x^{2}+16 y^{2}=144

, then

\mathrm{PA}+\mathrm{PB}

is equal to :
The slope of a common tangent to the ellipse

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

and a concentric circle of radius

r

, is A tangent to the ellipse

\frac{x^2}{25}+\frac{y^2}{16}=1

at any point

P

meet the line

x=0

at a point

Q

. Let

R

be the image of

Q

in the line

y=x

, then circle whose extremities of a diameter are

Q

and

R

passes through a fixed point. The fixed point is Any ordinate

N P

of an ellipse meets the auxiliary circle at

Q

. Then, the locus of the intersection of the normal at

P

and

O

is the circle