Chapter 19

If \(S=x^{2}+y^{2}+2 g x+2 f y+c=0\) is a given circle, then the locus of the foot of the perpendicular drawn from origin upon any chord of \(S\) which subtends a right angle at the origin, is (A) \(2\left(x^{2}+y^{2}\right)+2 g x+2 f y+c=0\) (B) \(2\left(x^{2}+y^{2}\right)+2 g x+2 f y-c=0\) (C) \(x^{2}+y^{2}+g x+f y+c=0\) (D) none of these

The equation of the circle, having the lines \(x^{2}+2 x y+\) \(3 x+6 y=0\) as its normals and having size just sufficient to contain the circle \(x(x-4)+y(y-3)=0\), is (A) \(x^{2}+y^{2}+6 x+3 y-45=0\) (B) \(x^{2}+y^{2}+6 x-3 y-45=0\) (C) \(x^{2}+y^{2}+6 x-3 y+45=0\) (D) none of these

The equation of the system of coaxal circles that are tangent at \((\sqrt{2}, 4)\) to the locus of the point of intersection of mutually \(\perp\) tangents to the circle \(x^{2}+y^{2}=9\), is (A) \(\left(x^{2}+y^{2}-18\right)+\lambda(\sqrt{2} x+4 y-18)=0\) (B) \(\left(x^{2}+y^{2}-18\right)+\lambda(4 x+\sqrt{2 y}-18)=0\) (C) \(\left(x^{2}+y^{2}-16\right)+\lambda(\sqrt{2 x}+4 y-16)=0\) (D) none of these

The point on the straight line \(y=2 x+11\) which is nearest to the circle \(16\left(x^{2}+y^{2}\right)+32 x-8 y-50=0\) is (A) \(\left(\frac{9}{2}, 2\right)\) (B) \(\left(-\frac{9}{2}, 2\right)\) (C) \(\left(\frac{9}{2},-2\right)\) (D) none of these

Extremities of a diagonal of a rectangle are \((0,0)\) and \((4,3)\). The equations of the tangents to the circumcircle of the rectangle which are parallel to this diagonal are (A) \(16 x+8 y \pm 25=0\) (B) \(6 x-8 y \pm 25=0\) (C) \(8 x+6 y \pm 25=0\) (D) none of these

The base \(A B\) of a triangle is fixed and its vertex \(C\) moves such that \(\sin A=k \sin B(k \neq 1)\). If \(a\) is the length of the base \(A B\), then the locus of \(C\) is a circle whose radius is equal to (A) \(\frac{a k}{\left(2-k^{2}\right)}\) (B) \(\frac{a k}{\left(1-k^{2}\right)}\) (C) \(\frac{2 a k}{1-k^{2}}\) (D) none of these

The locus of the centre of a circle touching the circle \(x^{2}+y^{2}-4 y-2 x=2 \sqrt{3}-1\) internally and tangents on which from \((1,2)\) is making a \(60^{\circ}\) angle with each other, is (A) \((x-1)^{2}+(y-2)^{2}=3\) (B) \((x-2)^{2}+(y-1)^{2}=1+2 \sqrt{3}\) (C) \(x^{2}+y^{2}=1\) (D) none of these

The equation of locus of the point of intersection of tangents to the circle \(x^{2}+y^{2}=1\) at the points whose parametric angles differ by \(60^{\circ}\) is (A) \(3 x^{2}+3 y^{2}=1\) (B) \(x^{2}+y^{2}=3\) (C) \(3 x^{2}+3 y^{2}=4\) (D) none of these

If a square is inscribed in the circle \(x^{2}+y^{2}-2 x+4 y+\) \(3=0\) and its sides are parallel to the coordinate axes, then one vertex of the square is (A) \((1+\sqrt{2},-2)\) (B) \((1-\sqrt{2},-2)\) (C) \((1,-2+\sqrt{2})\) (D) none of these

The equation of the chord of the circle \(x^{2}+y^{2}=a^{2}\) passing through the point \((2,3)\) and farthest from the centre is (A) \(2 x+3 y=13\) (B) \(3 x+2 y=13\) (C) \(2 x-3 y=13\) (D) none of these

The range of values of \(p\) such that the angle \(\theta\) between the pair of tangents drawn from the point \((p, 0)\) to the circle \(x^{2}+y^{2}=1\) lies in \(\left(\frac{\pi}{3}, \pi\right)\) is (A) \((-2,-1) \cup(1,2)\) (B) \((-3,-2) \cup(2,3)\) (C) \((0,2)\) (D) none of these

A circle whose centre coincides with the origin having radius ' \(a\) ' cuts \(x\)-axis at \(A\) and \(B\). If \(P\) and \(Q\) are two points on the circle whose parametric angles differ by \(2 \theta\), then the locus of the intersection point of \(A P\) and \(B Q\) is (A) \(x^{2}+y^{2}+2 a y \tan \theta=a^{2}\) (B) \(x^{2}+y^{2}-2 a y \tan \theta=a^{2}\) (C) \(x^{2}+y^{2}+2 a y \cot \theta=a^{2}\) (D) none of these

If a chord \(A B\) subtends a right angle at the centre of a given circle, then the locus of the centroid of the triangle \(P A B\) as \(P\) moves on the circle is a/an (A) parabola (B) ellipse (C) hyperbola (D) circle

If \(-3 R-6 l-1+6 m^{2}=0\) then the equation of the circle for which \(l x+m y+1=0\) is a tangent is (A) \((x+3)^{2}+y^{2}=6\) (B) \((x-3)^{2}+y^{2}=6\) (C) \(x^{2}+(y-3)^{2}=6\) (D) \(x^{2}+(y+3)^{2}=6\)

Let \(S_{1}\) and \(S_{2}\) be two circles with \(S_{2}\) lying inside \(S_{1}\). A circle \(S\) lying inside \(S_{1}\) touches \(S_{1}\) internally and \(\dot{S}_{2}\) externally. The locus of the centre of \(S\) is a/an (A) parabola (B) ellipse (C) hyperbola (D) circle

\(S(x, y)=0\) represents a circle. The equation \(S(x, 2)=\) 0 gives two identical solutions \(x=1\) and the equation \(S(1, y)=0\) gives two distinct solutions \(y=0,2\). The equation of the circle is (A) \(x^{2}+y^{2}+2 x+2 y+1=0\) (B) \(x^{2}+y^{2}+2 x+2 y-1=0\) (C) \(x^{2}+y^{2}-2 x-2 y+1=0\) (D) none of these

The equation of a circle of radius 2 touching the circles \(x^{2}+y^{2}-4|x|=0\) is (A) \(x^{2}+y^{2}+2 \sqrt{3} y+2=0\) (B) \(x^{2}+y^{2}+4 \sqrt{3} y+8=0\) (C) \(x^{2}+y^{2}-4 \sqrt{3} y+8=0\) (D) none of these

The coordinates of a point on the line \(y=2\) from which the tangents drawn to the circle \(x^{2}+y^{2}=25\) are perpendicular, are (A) \((\sqrt{46}, 2)\) (B) \((-\sqrt{46}, 2)\) (C) \((\sqrt{37}, 2)\) (D) \((-\sqrt{37}, 2)\)

Extremities of a diagonal of a rectangle are \((0,0)\) and (4, 3). The equations of the tangents to the circumcircle of the rectangle which are parallel to this diagonal are (A) \(16 x+8 y+25=0\) (B) \(6 x-8 y+25=0\) (C) \(8 x+6 y-25=0\) (D) \(6 x-8 y-25=0\)

The equation of the circle, touching the axis of \(x\) at the origin and the line \(3 y=4 x+24\), is (A) \(x^{2}+y^{2}+24 y=0\) (B) \(x^{2}+y^{2}-6 y=0\) (C) \(x^{2}+y^{2}-24 y=0\) (D) \(x^{2}+y^{2}+6 y=0\)

The coordinates of two points on the circle \(x^{2}+y^{2}-\) \(12 x-16 y+75=0\), one nearest to the origin and the other farthest from it, are (A) \((3,4)\) (B) \((3,2)\) (C) \((9,12)\) (D) \((9,-12)\)

The equation of a circle of equal radius, touching both the circles \(x^{2}+y^{2}=a^{2}\) and \((x-2 a)^{2}+y^{2}=a^{2}\) is given by (A) \(x^{2}+y^{2}-2 a x-2 \sqrt{3} a y+3 a^{2}=0\) (B) \(x^{2}+y^{2}-2 a x+2 \sqrt{3} a y+3 a^{2}=0\) (C) \(x^{2}+y^{2}+2 a x-2 \sqrt{3} a y+3 a^{2}=0\) (D) none of these

With respect to the circle \(x^{2}+y^{2}+6 x-8 y-10=0\) (A) The chord of contact of tangents from \((2,1)\) is \(5 x\) \(-3 y-8=0\) (B) the pole of the line \(5 x-3 y-8=0\) is \((2,1)\) (C) the polar of the point \((2,1)\) is \(5 x-3 y-8=0\) (D) all of these

For the circles \(S_{1}=x^{2}+y^{2}-4 x-6 y-12=0\) and \(S_{2}=\) \(x^{2}+y^{2}+6 x+4 y-12=0\) and the line \(L=x+y=0\) (A) \(L\) is the common tangent of \(S_{1}\) and \(S_{2}\) (B) \(L\) is the common chord of \(S_{1}\) and \(S_{2}\) (C) \(L\) is radical axis of \(S_{1}\) and \(S_{2}\) (D) \(L\) is perpendicular to the line joining the centres of \(S_{1}\) and \(S_{2}\)

Two circles, each of radius 5 units, touch each other at \((1,2)\). If the equation of their common tangent is \(4 x+\) \(3 y=10\), then the equations of the circles are (A) \(x^{2}+y^{2}+10 x+10 y+25=0\) (B) \(x^{2}+y^{2}-10 x-10 y+25=0\) (C) \(x^{2}+y^{2}+6 x+2 y-15=0\) (D) \(x^{2}+y^{2}-6 x-2 y+15=0\)

If the circle \(C_{1}: x^{2}+y^{2}=16\) intersects another circle \(C_{2}\) of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to \(\frac{3}{4}\), then the coordinates of the centre of \(C_{2}\) are (A) \(\left(\frac{9}{5},-\frac{12}{5}\right)\) (B) \(\left(-\frac{9}{5}, \frac{12}{5}\right)\) (C) \(\left(\frac{9}{5}, \frac{12}{5}\right)\) (D) \(\left(-\frac{9}{5},-\frac{12}{5}\right)\)

The circle \(x^{2}+y^{2}-4 x-4 y+4=0\) is inscribed in a triangle which have two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is \(x+y-x y+k \sqrt{x^{2}+y^{2}}=0\), then \(k\) is equal to (A) 1 (B) \(-1\) (C) 2 (D) none of these

From a point on the line \(4 x-3 y=6\), tangents are drawn to the circle \(x^{2}+y^{2}-6 x-4 y+4=0\) which make an angle of \(\tan -1 \frac{24}{7}\) between them, the coordinates of such points are (A) \((0,2)\) (B) \((0,-2)\) (C) \((6,6)\) (D) \((-6,6)\)

A tangent drawn from the point \((4,0)\) to the circle \(x^{2}+\) \(y^{2}=8\) touches it at a point \(A\) in the first quadrant. The coordinates of another point \(B\) on the circle such that \(A B=4\), are (A) \((2,-2)\) (B) \((-2,2)\) (C) \((2,2)\) (D) \((-2,-2)\)

Two vertices of an equilateral triangle are \((-1,0)\) and \((1,0)\). An equation of its circumcentre is (A) \(x^{2}+y^{2}+\frac{2}{\sqrt{3}} y-1=0\) (B) \(x^{2}+y^{2}-\frac{2}{\sqrt{3}} y-1=0\) (C) \(x^{2}+y^{2}+\frac{2}{\sqrt{3}} y+1=0\) (D) none of these

A tangent to the circle \(x^{2}+y^{2}=1\) through the point \((0,5)\) cuts the circle \(x^{2}+y^{2}=4\) at \(A\) and \(B\). The tangents for the circle \(x^{2}+y^{2}=4\) at \(A\) and \(B\) meet at \(C\). The coordinates of \(C\) are (A) \(\left(\frac{8 \sqrt{6}}{5}, \frac{4}{5}\right)\) (B) \(\left(-\frac{8 \sqrt{6}}{5}, \frac{4}{5}\right)\) (C) \(\left(\frac{8 \sqrt{6}}{5},-\frac{4}{5}\right)\) (D) \(\left(-\frac{8 \sqrt{6}}{5},-\frac{4}{5}\right)\)

The equation \(r=|\cos \theta|\) represents (A) two circles of radii \(\frac{1}{2}\) each (B) two circles centered at \(\left(\frac{1}{2}, 0\right)\) and \(\left(-\frac{1}{2}, 0\right)\) (C) two circles touching each other at the origin (D) pair of straight lines

If the polar of \(P\) with respect to the circle \(x^{2}+y^{2}=a^{2}\) touches the circle \((x-f)^{2}+(y-g)^{2}=b^{2}\), then its locus is given by the equation (A) \(\left(f x+g y-a^{2}\right)^{2}=a^{2}\left(x^{2}+y^{2}\right)\) (B) \(\left(f x+g y-a^{2}\right)^{2}=b^{2}\left(x^{2}+y^{2}\right)\) (C) \(\left(f x-g y-a^{2}\right)^{2}=a^{2}\left(x^{2}+y^{2}\right)\) (D) none of these

The pole of the line \(3 x+4 y=45\) with respect to the circle \(x^{2}+y^{2}-6 x-8 y+5=0\) is (A) \((6,8)\) (B) \((6,-8)\) (C) \((-6,8)\) (D) \((-6,-8)\)

The pole of the chord of the circle \(x^{2}+y^{2}=16\) which is bisected at the point \((-2,3)\), with respect to the circle is (A) \(\left(\frac{-32}{13}, \frac{48}{13}\right)\) (B) \(\left(\frac{32}{13}, \frac{48}{13}\right)\) (C) \(\left(\frac{-32}{13}, \frac{-48}{13}\right)\) (D) none of these

The coordinates of the poles of the common chord of the circles \(x^{2}+y^{2}=12\) and \(x^{2}+y^{2}-5 x+2 y-2=0\) with respect to the circle \(x^{2}+y^{2}=12\) are (A) \(\left(6, \frac{-12}{5}\right)\) (B) \(\left(-6, \frac{12}{5}\right)\) (C) \(\left(6, \frac{12}{5}\right)\) (D) none of these

If \(A, B, C\) be the centres and \(r_{1}, r_{2}, r_{3}\) the radii of three coaxal circles, then \(r_{1}^{2} \cdot B C+r_{2}^{2} \cdot C A+r_{2}^{3} \cdot A B=\) (A) \(B C \cdot C A \cdot A B\) (B) \(-B C \cdot C A \cdot A B\) (C) \(2 B C \cdot C A \cdot A B\) (D) none of these

The limiting points of the coaxal system determined by the circles \(x^{2}+y^{2}-2 x-6 y+9=0\) and \(x^{2}+y^{2}+6 x\) \(-2 y+1=0\) are (A) \((-1,2),\left(\frac{3}{5}, \frac{-14}{5}\right)\) (B) \((-1,2),\left(\frac{3}{5}, \frac{14}{5}\right)\) (C) \((-1,2),\left(\frac{-3}{5}, \frac{14}{5}\right)\) (D) none of these

The equation of the circle which passes through the origin and belongs to the coaxal system whose limiting points are \((1,2)\) and \((4,3)\), is (A) \(2 x^{2}+2 y^{2}-x-7 y=0\) (B) \(2 x^{2}+2 y^{2}+x-7 y=0\) (C) \(2 x^{2}+2 y^{2}+x+7 y=0\) (D) none of these

I. The number of common tangents to (A) 3 the circles \(x^{2}+y^{2}-6 x-2 y+9=0\) and \(x^{2}+y^{2}-14 x-8 y+61=0\) is II. The number of common tangents to (B) 4 the circles \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}-8 x\) \(+12=0\) is III. The number of common tangents to (C) 2 the circles \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}-6 x\) \(-8 y-24=0\) is IV. The number of tangents to the circle (D) 1 \(x^{2}+y^{2}-8 x-6 y+9=0\) which pass through the point \((3,-2)\) is

Assertion: The locus of the centres of circles passing through the origin and cutting the circle \(x^{2}+y^{2}+6 x-\) \(4 y+2=0\) orthogonally is \(3 x-2 y+1=0\). Reason: The two circles \(x^{2}+y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0\) and \(x^{2}+y^{2}+2 g_{2} x+2 f_{2} y+c_{2}=0\) cut each other orthogonally if \(2 g_{1} g_{2}+2 f f_{2}=c_{1}+c_{2}\)

Assertion: The tangent to the circle \(x^{2}+y^{2}=5\) at the point \((1,-2)\) also touches the circle \(x^{2}+y^{2}-8 x+6 y+\) \(20=0\). Then its point of contact is \((3,-1)\). Reason: The equation of tangent to the circle \(x^{2}+y^{2}+\) \(2 g x+2 f y+c=0\) at the point \(\left(x_{1}, y_{1}\right)\) is \(x x_{1}+y y_{1}+g(x\) \(\left.+x_{1}\right)+f\left(y+y_{1}\right)+c=0 .\)

Assertion: If the point \((2,4)\) is interior to the circle \(x^{2}\) \(+y^{2}-6 x-10 y+k=0\) and the circle does not cut the axes at any point, then \(25

Assertion: The equation of the circle passing through the point \((2 a, 0)\) and whose radical axis is \(x=\frac{a}{2}\) with respect to the circle \(x^{2}+y^{2}=a^{2}\), will be \(x^{2}+y^{2}-2 a x=0\). Reason: The equation of radical axis of two circles \(x^{2}\) \(+y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0\) and \(x^{2}+y^{2}+2 g_{2} x+2 f y+c_{2}\) \(=0\) is \(2\left(g_{1}-g_{2}\right) x+2\left(f_{1}-f_{2}\right) y+\left(c_{1}-c_{2}\right)=0\).

The greatest distance of the point \(P(10,7)\) from the circle \(x^{2}+y^{2}-4 x-2 y-20=0\) is [2002| (A) 10 unit (B) 15 unit (C) 5 unit (D) none of these

The equation of the tangent to the circle \(x^{2}+y^{2}+4 x-\) \(4 y+4=0\) which make equal intercepts on the positive co-ordinate axes, is (A) \(x+y=2\) (B) \(x+y=2 \sqrt{2}\) (C) \(x+y=4\) (D) \(x+y=8\)

If the two circles \((x-1)^{2}+(y-3)^{2}=r^{2}\) and \(x^{2}+y^{2}\) \(-8 x+2 y+8=0\) intersect in two distinct points, then [2003] (A) \(22\)

The lines \(2 x-3 y=5\) and \(3 x-4 y=7\) are diameters of a circle having area as 154 sq units. Then the equation of the circle is (A) \(x^{2}+y^{2}+2 x-2 y=62\) (B) \(x^{2}+y^{2}+2 x-2 y=47\) (C) \(x^{2}+y^{2}-2 x+2 y=47\) (D) \(x^{2}+y^{2}-2 x+2 y=62\)

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

A variable circle passes through the fixed point \(A(p,\), \(q\) ) and touches \(x\)-axis. The locus of the other end of the diameter through \(A\) is (A) \((x-p)^{2}=4 q y\) (B) \((x-q)^{2}=4 p y\) (C) \((y-p)^{2}=4 q x\) (D) \((y-q)^{2}=4 p x\)