Chapter 19

If the lines \(2 x+3 y+1=0\) and \(3 x-y-4=0\) lie along diameters of a circle of circumference \(10 \pi\), then the equation of the circle is [2004] (A) \(x^{2}+y^{2}-2 x+2 y-23=0\) (B) \(x^{2}+y^{2}-2 x-2 y-23=0\) (C) \(x^{2}+y^{2}+2 x+2 y-23=0\) (D) \(x^{2}+y^{2}+2 x-2 y-23=0\)

The intercept on the line \(y=x\) by the circle \(x^{2}+y^{2}-2 x\) \(=0\) is \(A B\). Equation of the circle on \(A B\) as a diameter is \(|2004|\) (A) \(x^{2}+y^{2}-x-y=0\) (B) \(x^{2}+y^{2}-x+y=0\) (C) \(x^{2}+y^{2}+x+y=0\) (D) \(x^{2}+y^{2}+x-y=0\)

A circle touches the \(x\)-axis and also touches the circle with centre at \((0,3)\) and radius \(2 .\) The locus of the centre of the circle is (A) an cllipse (B) a circle (C) a hyperbola (D) a parabola

If a circle passes through the point \((a, b)\) and cuts the circle \(x^{2}+y^{2}=p^{2}\) orthogonally, then the equation of the locus of its centre is |2005] (A) \(x^{2}+y^{2}-3 a x-4 b y+\left(a^{2}+b^{2}-p^{2}\right)=0\) (B) \(2 a x+2 b y-\left(a^{2}-\beta^{2}+p^{2}\right)=0\) (C) \(x^{2}+y^{2}-2 a x-3 b y-\left(a^{2}-\beta^{2}-p^{2}\right)=0\) (D) \(2 a x+2 b y-\left(a^{2}+b^{2}+p^{2}\right)=0\)

If the lines \(3 x-4 y-7=0\) and \(2 x-3 y-5=0\) are two diameters of a circle of area \(49 \pi\) square units, the equation of the circle is \([2006]\) (A) \(x^{2}+y^{2}+2 x-2 y-47=0\) (B) \(x^{2}+y^{2}+2 x-2 y-62=0\) (C) \(x^{2}+y^{2}-2 x+2 y-62=0\) (D) \(x^{2}+y^{2}-2 x+2 y-47=0\)

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 (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 |2008| (A) \((3,-4)\) (B) \((-3,4)\) (C) \((-3,-4)\) (D) \((3,4)\)

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 \([\mathbf{2 0 1 3}]\) (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 [2014] (A) \(\begin{array}{ll}\frac{\sqrt{3}}{\sqrt{2}} & \text { (B) } \frac{\sqrt{3}}{2}\end{array}\) (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