Chapter 19

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 \(\frac{2 \pi}{3}\) at its centre is (A) \(x^{2}+y^{2}=\frac{3}{2}\) (B) \(x^{2}+y^{2}=1\) (C) \(x^{2}+y^{2}=\frac{27}{4}\) (D) \(x^{2}+y^{2}=\frac{9}{4}\)

Consider a family of circles which are passing through the point \((-1,1)\) and are tangent to \(x\)-axis. If \((h, k)\) are the co-ordinates of the centre of the circles, then the set of values of \(k\) is given by the interval [2007] (A) \(0

The point diametrically opposite to the point \(P(1,0)\) on the circle \(x^{2}+y^{2}+2 x+4 y-3=0\) is \(\quad\) [2008] (A) \((3,-4)\) (B) \((-3,4)\) (C) \((-3,-4)\) (D) \((3,4)\)

If \(P\) and \(Q\) are the points of intersection of the circles \(x^{2}+y^{2}+3 x+7 y+2 p-5=0\) and \(x^{2}+y^{2}+2 x+2 y-p^{2}\) \(=0\), then there is a circle passing through \(P, Q\) and (1, 1) for (A) all values of \(p\) (B) all except one value of \(p\) (C) all except two values of \(p\) (D) exactly one value of \(p\)

The circle \(x^{2}+y^{2}=4 x+8 y+5\) intersects the line \(3 x\) \(-4 y=m\) at two distinct points then \(\mathrm{m}\) satisfies [2010] (A) \(-35

The two circles \(x^{2}+y^{2}=a x\) and \(x^{2}+y^{2}=c^{2}(c>0)\) touch each other if (A) \(|a|=c\) (B) \(a=2 c\) (C) \(|a|=2 c\) (D) \(2|a|=c\)

The length of the diameter of the circle which touches the \(x\)-axis at the point \((1,0)\) and passes through the point \((2,3)\) is \([2012]\) (A) \(\frac{10}{3}\) (B) \(\frac{3}{5}\) (C) \(\frac{6}{5}\) (D) \(\frac{5}{3}\)

The circle passing through \((1,-2)\) and touching the \(x\)-axis at \((3,0)\) also passes through the point [2013] (A) \((2,-5)\) (B) \((5,-2)\) (C) \((-2,5)\) (D) \((-5,2)\)

Let \(C\) be the circle with centre at \((1,1)\) and with radius 1\. If \(T\) is the circle centered at \((0, y)\), passing through origin and touching the circle \(C\) externally, then the radius of \(T\) is equal to (A) \(\frac{\sqrt{3}}{\sqrt{2}}\) (B) \(\frac{\sqrt{3}}{2}\) (C) \(\frac{1}{2}\) (D) \(\frac{1}{4}\)

The number of common tangents to the circles \(x^{2}+y^{2}\) \(-4 x-6 y-12=0\) and \(x^{2}+y^{2}+6 x+18 y+26=0\), is \([2015]\) (A) 2 (B) 3 (C) 4 (D) 1

If one of the diameters of the circle, given by the equation. \(x^{2}+y^{2}-4 x+6 y-12=0\), is a chord of a circle \(S\), whose centre is at \((-3,2)\), then the radius of \(S\) is (A) 10 (B) \(5 \sqrt{2}\) (C) \(5 \sqrt{3}\) (D) 5