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(A) Consider the two curves
\( C_{1}: y^{2}=4 x ; \quad C_{2}: x^{2}+y^{2}-6 x+1=0 \). Then:
(a) \( C_{1} \) and \( C_{2} \) touch each other only at one point
(b) \( C_{1} \) and \( C_{2} \) touch each other exactly at two points
(c) \( C_{1} \) and \( C_{2} \) intersect (but do not touch) at exactly two points
(d) \( C_{1} \) and \( C_{2} \) neither intersect nor touch each other
(B) Consider, \( L_{1}: 2 x+3 y+P-3=0 ; \quad L_{2}: 2 x+3 y+P+3=0 \),
where \( P \) is a real number and \( C: x^{2}+y^{2}+6 x-10 y+30=0 \).
Statement-1: If line \( L_{1} \) is a diameter of circle \( C \), then line \( L_{2} \) is not always a diameter of circle \( C \)
because
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 explanation for statement-1.
(b) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.
I \( \alpha m \operatorname{Pr} \)
C) Comprehension
A circle \( C \) of radius 1 is inscribed in an equilateral triangle \( P Q R \). The points of contact of \( C \) with the sides \( P Q, Q R, R P \) are \( D, E, F \) respectively. The line \( P Q \) is given by the equation \( \sqrt{3} x+y-6=0 \) and the point \( D \) is \( \left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right) \).
Further, it is given that the origin and the centre of \( C \) are on the same side of the line \( P Q \).
[IIT-JEE 2008]
(i) The equation of circle \( C \) is:
(a) \( (x-2 \sqrt{3})^{2}+(y-1)^{2}=1 \)
(b) \( (x-2 \sqrt{3})^{2}+\left(y+\frac{1}{2}\right)^{2}=1 \)
(c) \( (x-\sqrt{3})^{2}+(y+1)^{2}=1 \)
(d) \( (x-\sqrt{3})^{2}+(y-1)^{2}=1 \)
(ii) Points \( E \) and \( F \) are given by:
(a) \( \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),(\sqrt{3}, 0) \)
(b) \( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right),(\sqrt{3}, 0) \)
(c) \( \left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right),\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \)
(d) \( \left(\frac{3}{2}, \frac{\sqrt{3}}{2}\right),\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) \)
(iii) Equations of the sides \( R P, R Q \) are:
(a) \( y=\frac{2}{\sqrt{3}} x+1, y=-\frac{2}{\sqrt{3}} x-1 \)
(b) \( y=\frac{1}{\sqrt{3}} x, y=0 \)
(c) \( y=\frac{\sqrt{3}}{2} x+1, y=-\frac{\sqrt{3}}{2} x-1 \)
(d) \( y=\sqrt{3} x, y=0 \)
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