Let and be the chords of passing through the point and parallel to the -axis and the -axis, respectively. Let be the chord of passing through and having slope . Let the tangents to at and meet at , the tangents to at and meet at , and the tan

Question

Let $S$ be the circle in the $x y$ -plane defined by the equation $x^{2} + y^{2} = 4$ . [2018]

Q. 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

Smart Achievers · Accepted Answer

(1)

$Equation of E_{1} E_{2} is y = 1; Equation of F_{1} F_{2} is x = 1$

$Equation of G_{1} G_{2} is x + y = 2$

$By symmetry, tangents at E_{1} and E_{2} meet on y-axis and tangents at F_{1} and F_{2} meet on x-axis.$

$E_{1} \equiv (\sqrt{3}, 1), F_{1} \equiv (1, \sqrt{3})$

$Equation of tangent at E_{1} is$ $\sqrt{3} x + y = 4$

$Equation of tangent at F_{1} is$ $x + \sqrt{3} y = 4$

$G_{3}$ $Points E_{3} (0, 4) and F_{3} (4, 0)$

$Tangents at G_{1} and G_{2} are x = 2 and y = 2 respectively, intersecting each other at G_{3} (2, 2)$

$Clearly E_{3}, F_{3} and G_{3} lie on the curve x + y = 4 .$

Solution

1 $x + y = 4$

2 ${(x - 4)}^{2} + {(y - 4)}^{2} = 16$

3 $(x - 4) (y - 4) = 4$

4 $x y = 4$

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Solution

1 x+y=4

2 (x-4)2+(y-4)2=16

3 (x-4)(y-4)=4

4 xy=4

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1 $x + y = 4$

2 ${(x - 4)}^{2} + {(y - 4)}^{2} = 16$

3 $(x - 4) (y - 4) = 4$

4 $x y = 4$