The point diametrically opposite to the point $$P(1, 0)$$ on the circle $${x^2} + {y^2} + 2x + 4y - 3 = 0$$ is
A
$$(3, -4)$$
B
$$(-3, 4)$$
C
$$(-3, -4)$$
D
$$(3, 4)$$
Explanation
The given circle is $${x^2} + {y^2} + 2x + 4y - 3 = 0$$
Center $$(-1,-2)$$
Let $$Q$$ $$\left( {\alpha ,\beta } \right)$$ be the point diametrically opposite to the point $$P(1,0),$$
then $${{1 + \alpha } \over 2} = - 1$$ and $${{0 + \beta } \over 2} = - 2$$
$$ \Rightarrow \alpha = - 3,\beta = - 4,$$ So, $$Q$$ is $$\left( { - 3, - 4} \right)$$
3
AIEEE 2007
MCQ (Single Correct Answer)
Consider a family of circles which are passing through the point $$(-1, 1)$$ and are tangent to $$x$$-axis. If $$(h, k)$$ are the coordinate of the centre of the circles, then the set of values of $$k$$ is given by the interval
A
$$ - {1 \over 2} \le k \le {1 \over 2}$$
B
$$k \le {1 \over 2}$$
C
$$0 \le k \le {1 \over 2}$$
D
$$k \ge {1 \over 2}$$
Explanation
Equation of circle whose center is $$\left( {h,k} \right)$$
Let $$C$$ be the circle with centre $$(0, 0)$$ and radius $$3$$ units. The equation of the locus of the mid points of the chords of the circle $$C$$ that subtend an angle of $${{2\pi } \over 3}$$ at its center is
A
$${x^2} + {y^2} = {3 \over 2}$$
B
$${x^2} + {y^2} = 1$$
C
$${x^2} + {y^2} = {{27} \over 4}$$
D
$${x^2} + {y^2} = {{9} \over 4}$$
Explanation
Let $$M\left( {h,k} \right)$$ be the mid point of chord $$AB$$ where