Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.
The radius of the circumcircle of the triangle PRS is
Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.
The radius of the incircle of the triangle PQR is
Consider the following linear equations
$$ax + by + cz = 0$$
$$bx + cy + az = 0$$
$$cx + ay + bz = 0$$
Match the conditions/expressions in Column I with statements in Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | $$a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$ | (P) | the equations represent planes meeting only at a single point. |
| (B) | $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$ | (Q) | the equations represent the line $$x=y=z$$. |
| (C) | $$a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$ | (R) | the equations represent identical planes. |
| (D) | $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$ | (S) | the equations represent the whole of the three dimensional space. |
In the following [x] denotes the greatest integer less than or equal to x.
Match the functions in Column I with the properties Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | $$x|x|$$ | (P) | continuous in ($$-1,1$$). |
| (B) | $$\sqrt{|x|}$$ | (Q) | differentiable in ($$-1,1$$) |
| (C) | $$x+[x]$$ | (R) | strictly increasing in ($$-1,1$$) |
| (D) | $$|x-1|+|x+1|$$ | (S) | not differentiable at least at one point in ($$-1,1$$) |
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