Question

PARAGRAPH "X"
Let $S$ be the circle in the $xy$-plane defined by the equation $x^2 + y^2 = 4$.
Question: Let $E_1E_2$ and $F_1F_2$ 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 $s$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3$, and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3 . F_3$, and $G_3$ lie on the curve

• A. $x +y = 4 $
• B. $(x - 4)^2 + (y - 4)^2 = 16 $
• C. $(x - 4 )(y - 4) = 4 $
• D. $x y = 4 $

Answer 1

Answer: (A)

Co-ordinate of $E_{1}$ and $E_{2}$ are obtained
by solving $y=1$ and $x^{2}+y^{2}=4$
$\therefore E_{1}(-\sqrt{3}, 1)$ and $E_{2}(-\sqrt{3}, 1)$
co-ordinates of $F_{1}$ and $F_{2}$ are obtained
by solving $x=1$ and $x^{2}+y^{2}=4$
$F_{1}(1, \sqrt{3})$ and $F_{2}(1,-\sqrt{3}) $
Tangent at $E_{1}:-\sqrt{3} x+y=4$
Tangent at $E_{2}:-\sqrt{3} x+y=4 $
$ \therefore E_{3}(0,4)$
Tangent at $F_{1}: x+\sqrt{3} y=4 $
Tangent at $F_{2}: x-\sqrt{3} y=4 $
$ \therefore F_{3}(4,0)$
And similarly $G_{3}(2,2) (0,4),(4,0)$ and $(2,2)$
lies on $x+y=4$