JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Circle

Passage

Let

S

be the circle in the

x y

-plane defined by the equation

x^{2} + y^{2} = 4

Let

S

be the circle in the

x y

-plane defined by the equation

x^{2} + y^{2} = 4

.

Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chords of

S

passing through the point

P_{0} (1, 1)

and parallel to the

x

-axis and the

y

-axis, respectively. Let

G_{1} G_{2}

be the chord of

S

passing through

P_{0}

and having slope -1. Let the tangents to

S

E_{1}

and

E_{2}

meet at

E_{3}

, the tangents to

S

F_{1}

and

F_{2}

meet at

F_{3}

, and the tangents to

S

G_{1}

and

G_{2}

meet at

G_{3}

. Then, the points

E_{3}, F_{3}

, and

G_{3}

lie on the curve

Answer: (A)

Solution

For a chord of

x^{2} + y^{2} = 4

passing through

P_{0} (1, 1)

, the locus of the intersection of tangents at the endpoints is the polar of

P_{0}

, which is

1 \cdot x + 1 \cdot y = 4

, i.e.,

x + y = 4

.

This applies to all three chords, so

E_{3}, F_{3}, G_{3}

all lie on

x + y = 4

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Single correct option