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1

### IIT-JEE 2012 Paper 1 Offline

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle $${x^2}\, + \,{y^2} = 9$$ is
A
$$20\,({x^2}\, + \,{y^2}) - \,\,36x\,\, + \,\,45y = 0$$
B
$$20\,({x^2}\, + \,{y^2}) + \,\,36x\,\, - \,\,45y = 0$$
C
$$36\,({x^2}\, + \,{y^2}) - \,\,20x\,\, + \,\,45y = 0$$
D
$$36\,({x^2}\, + \,{y^2}) + \,\,20x\,\, - \,\,45y = 0$$
2

### IIT-JEE 2011 Paper 2 Offline

The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point.
A
$$\left( { - {3 \over 0},0} \right)$$
B
$$\left( { - {5 \over 2},2} \right)$$
C
$$\left( { - {3 \over 0},\,{5 \over 2}} \right)$$
D
(- 4, 0)
3

### IIT-JEE 2009

Tangents drawn from the point P (1, 8) to the circle
$${x^2}\, + \,{y^2}\, - \,6x\, - 4y\, - 11 = 0$$
touch the circle at the points A and B. The equation of the cirumcircle of the triangle PAB is
A
$${x^2}\, + \,{y^2}\, + \,4x\,\, - 6y\, + 19 = 0$$
B
$${x^2}\, + \,{y^2}\, - \,4x\,\, - 10y\, + 19 = 0$$
C
$${x^2}\, + \,{y^2}\, - \,2x\,\, + 6y\, - 29 = 0$$
D
$${x^2}\, + \,{y^2}\, - \,6x\,\, - 4y\, + 19 = 0$$
4

### IIT-JEE 2008

Consider $$\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0$$
$$\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0$$

where p is a real number, and $$\,C:\,{x^2}\, + \,{y^2}\, + \,6x\, - 10y\, + \,30 = 0$$
STATEMENT-1 : If line $${L_1}$$ is a chord of circle C, then line $${L_2}$$ is not always a diameter of circle C
and
STATEMENT-2 : If line $${L_1}$$ is a diameter of circle C, then line $${L_2}$$ is not a chord of circle C.

A
Statement-1 is True, Statement-2 is True; Statement-2 is a correct rexplanation for Statement-1
B
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct rexplanation for Statement-1
C
Statement-1 is True, Statement-2 is False
D
Statement-1 is False, Statement-2 is True

## Explanation

Equation of circle C is

$${(x + 3)^2} + {(y - 5)^2} = 9 + 25 - 30 = 4$$

$$\Rightarrow {(x + 3)^2} + {(y - 5)^2} = {2^2}$$

Centre = (3, $$-$$5)

If L1 is diameter, then $$2(3) + 3( - 5) + p - 3 = 0 \Rightarrow p = 12$$

$$\therefore$$ L1 is $$2x + 3y + 9 = 0$$

and L2 is $$2x + 3y + 15 = 0$$

Distance of centre of circle from $${L_2} = \left| {{{2(3) + 3( - 5) + 15} \over {\sqrt {{2^2} + {3^2}} }}} \right| = {6 \over {\sqrt {12} }} < 2$$ [radius of circle]

$$\therefore$$ L2 is a chord of circle C.

Statement 2 is false.

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