JEE Advanced 2020 Concept Recapitulation Test 1 Paper 1

Let P(a‌cos‌θ1,b‌sin‌θ1),Q(a‌cos‌θ2,b‌sin‌θ2) and R(a‌cos‌θ3,b‌sin‌θ3) be the vertices of the triangle inscribed in the ellipse ‌

x2

a2

+‌

y2

b2

=1.
The points on the auxiliary circle corresponding tothese points are P′(a‌cos‌θ1,a‌sin‌θ1),Q′(a‌cos‌θ2,a‌sin‌θ2) and R′(a‌cos‌θ3,a‌sin‌θ3)
∴∆1= Area of ∆PQR
=‌

1

2

|

cos‌θ1	b‌sin‌θ1	1
a‌cos‌θ2	b‌sin‌θ2	1
a‌cos‌θ3	b‌sin‌θ3	1

| =‌

1

2

ab|

cos‌θ1	sin‌θ1	1
cos‌θ2	sin‌θ2	1
cos‌θ3	sin‌θ3	1

| and ∆2= Area of ∆P′Q′R′ =‌

1

2

|

a‌cos‌θ1	a‌sin‌θ1	1
a‌cos‌θ2	a‌sin‌θ2	1
a‌cos‌θ3	a‌sin‌θ3	1

|
=‌

1

2

a2|

cos‌θ1	sin‌θ1	1
cos‌θ2	sin‌θ2	1
cos‌θ3	sin‌θ3	1

|
Clearly, ‌

∆1

∆2

=‌

1

7

⇒‌‌‌

∆2

∆1

=7