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Question

Let S be the circle in the xy-plane defined by the equation x2+y2=4.

Let E1E2 and F1F2 be the chords of S passing through the point P0(1,1) and parallel to the x-axis and the y-axis, respectively. Let G1G2 be the chord of S passing through P0 and having slope 1. Let the tangents to S at E1 and E2 meet at E3, the tangents to S at F1 and F2 meet at F3, and the tangents to S at G1 and G2 meet at G3. Then, the points E3,F3, and G3 lie on the curve

A
x+y=4
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B
(x4)2+(y4)2=16
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C
(x4)(y4)=4
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D
xy=4
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Solution

The correct option is A x+y=4
The equation of circle: x2+y2=4

Coordinates of points,
E1=(3,1)E2=(3,1)
Tangents at E1 and E2,
3x+y=4..........(i)3x+y=4......(ii)
Using equation (i) and (ii),
E3=(0,4)
Similarly,
F1=(1,3)F2=(1,3)
Tangents at F1 and F2,
x+3y=4..........(iii)x3y=4......(iv)
Using equation (iii) and (iv),
F3=(4,0)
Now for G3,
G1=(2,0)G2=(0,2)
Tangents at G1 and G2,
2x=4..........(v)2y=4......(vi)
Using equation (v) and (vi),
G3=(2,2)
Hence the points E3=(0,4),F3=(4,0),G3=(2,2) lies on
x+y=4

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