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PARAGRAPH "X" Let S be the circle in th...

PARAGRAPH `"X"` Let `S` be the circle in the `x y` -plane defined by the equation `x^2+y^2=4.` (For Ques. No 15 and 16) 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_1G_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 `x+y=4` (b) `(x-4)^2+(y-4)^2=16` (c) `(x-4)(y-4)=4` (d) `x y=4`

A

`x+y=4`

B

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

C

`(x-4)(y-4)=4`

D

`xy=4`

Text Solution

Verified by Experts

The correct Answer is:
A

Circle is `x^(2)+y^(2) =4`.
Equation of chord through `P_(0) (1,1)` parallel to x-axis is `y=1`
Solving this with circle, we `E_(1)(sqrt(3),1)` and `E_(2)(- sqrt(3),1)`.
Equations of tangent to circle at `E_(1)` and `E_(2)` are `sqrt(3)x+y=4` and `- sqrt(3) x +y=4` , respectively.
These tangents meet at `E_(3)(0,4)`.
Chord through `p_(0)(1,1)`, parallel to y -axis is `x=1`, which meets circle at `F_(1)(1,sqrt(3))` and `F_(2) (1,-sqrt(3))`.
Tangents to circle at `F_(1)` and `F_(2)` meet at `F_(3)(0,4)`.
Chord through `P_(0)` having slope -1 is `x+y=2` which meets circle at `G_(1)(0,2)` and `G_(2)(2,0)`.
Tangest to circle at `G_(1)` and `G_(2)` meet at `G_(3)(2,2)`.
Clearly, `E_(3),F_(3)` and `G_(3)` lie on line `x+y=4`.
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