Question

PARAGRAPH "X"
Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.
Question: 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 $s$ 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. $xy=4$

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 $x+y=4$