JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Ellipse

Consider ellipses

E_{k} : k x^{2} + k^{2} y^{2} = 1, k = 1, 2, \dots, 20

. Let

C_{k}

be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse

E_{k}

. If

r_{k}

is the radius of the circle

C_{k}

, then the value of

\sum_{k = 1}^{20} \frac{1}{r _{k}^{2}}

Answer: (A)

Solution

The ellipse

E_{k}

can be written as

\frac{x ^{2}}{1/ k} + \frac{y ^{2}}{1/ k ^{2}} = 1

. The endpoints are

A_{1} (1/ k, 0)

and

B_{1} (0, 1/ k)

. The line

A_{1} B_{1}

has equation

k x + k y = 1

. The radius

r_{k}

is the perpendicular distance from origin:

r_{k} = \frac{1}{k + k ^{2}}

. So

\frac{1}{r _{k}^{2}} = k + k^{2}

. Therefore

\sum_{k = 1}^{20} \frac{1}{r _{k}^{2}} = \sum_{k = 1}^{20} (k + k^{2}) = \frac{20 \cdot 21}{2} + \frac{20 \cdot 21 \cdot 41}{6} = 210 + 2870 = 3080

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option