Chapter 11

If the length of the chord of the circle, \(x^{2}+y^{2}=r^{2}(r>0)\) along the line, \(y-2 x=3\) is \(r\), then \(r^{2}\) is equal to: (a) \(\frac{9}{5}\) (b) 12 (c) \(\frac{24}{5}\) (d) \(\frac{12}{5}\)

Let \(P Q\) be a diameter of the circle \(x^{2}+y^{2}=9\). If \(\alpha\) and \(\beta\) are the lengths of the perpendiculars from \(P\) and \(Q\) on the straight line, \(x+y=2\) respectively, then the maximum value of \(\alpha \beta\) is ___ .

The diameter of the circle, whose centre lies on the line \(x+y=2\) in the first quadrant and which touches both the lines \(x=3\) and \(y=2\), is ____ .

The number of integral values of \(k\) for which the line, \(3 x+4 y=k\) intersects the circle, \(x^{2}+y^{2}-2 x-4 y+4=0\) at two distinct points is ____ .

If the curves, \(x^{2}-6 x+y^{2}+8=0\) and \(x^{2}-8 y+y^{2}+16-k=0\), \((k>0)\) touch each other at a point, then the largest value of \(k\) is _____ .

Let the tangents drawn from the origin to the circle, \(x^{2}+y^{2}-8 x-4 y+16=0\) touch it at the points \(A\) and \(B\). The \(\begin{array}{ll}(A B)^{2} \text { is equal to: (a) \)\frac{52}{5}\( (b) \)\frac{56}{5}\( (c) \)\frac{64}{5}\( (d) \)\frac{32}{5}$

If the angle of intersection at a point where the two circles with radii \(5 \mathrm{~cm}\) and \(12 \mathrm{~cm}\) intersect is \(90^{\circ}\), then the length (in \(\mathrm{cm}\) ) of their common chord is : (a) \(\frac{13}{5}\) (b) \(\frac{120}{13}\) (c) \(\frac{60}{13}\) (d) \(\frac{13}{2}\)

A circle touching the \(x\)-axis at \((3,0)\) and making an intercept of length 8 on the \(y\)-axis passes through the point (a) \(\quad(3,10)\) (b) \((3,5)\) (c) \((2,3)\) (d) \((1,5)\)

If the circles \(x^{2}+y^{2}+5 K x+2 y+K=0\) and \(2\left(x^{2}+y^{2}\right)+\) \(2 \mathrm{Kx}+3 \mathrm{y}-1=0,(\mathrm{~K} \in \mathbf{R})\), intersect at the points \(\mathrm{P}\) and \(\mathrm{Q}\), then the line \(4 x+5 y-K=0\) passes through \(P\) and \(Q\), for: (a) infinitely many values of \(\mathrm{K}\) (b) no value of \(\mathrm{K}\). (c) exactly two values of \(\mathrm{K}\) (d) exactly one value of \(\mathrm{K}\)

The line \(\mathrm{x}=\mathrm{y}\) touches a circle at the point \((1,1)\). If the circle also passes through the point \((1,-3)\), then its radius is: (a) 3 (b) \(2 \sqrt{2}\) (c) 2 (d) \(3 \sqrt{2}\)

The locus of the centres of the circles, which touch the circle, \(x^{2}+y^{2}=1\) externally, also touch the \(y\)-axis and lie in the first quadrant, is: (a) \(x=\sqrt{1+4 y}, y \geq 0\) (b) \(y=\sqrt{1+2 x}, x \geq 0\) (c) \(y=\sqrt{1+4 x}, x \geq 0\) (d) \(x=\sqrt{1+2 y}, y \geq 0\)

All the points in the set \(\mathbf{S}=\left\\{\frac{\alpha+\mathrm{i}}{\alpha-1} ; \propto \in \mathrm{R}\right\\}(\mathrm{i}=\sqrt{-1})\) lie on a: (a) straight line whose slope is 1 . (b) circle whose radius is 1 . (c) circle whose radius is \(\sqrt{2}\). (d) straight line whose slope is \(-1\).

The common tangent to the circles \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}+\) \(6 x+8 y-24=0\) also passes through the point: (a) \((4,-2)\) (b) \((-6,4)\) (c) \((6,-2)\) (d) \((-4,6)\)

The sum of the squares of the lengths of the chords intercepted on the circle, \(x^{2}+y^{2}=16\), by the lines, \(x+y=n\), \(\mathrm{n} \in \mathrm{N}\), where \(\mathrm{N}\) is the set of all natural numbers, is: (a) 320 (b) 105 (c) 160 (d) 210

If a circle of radius \(\mathrm{R}\) passes through the origin \(\mathrm{O}\) and intersects the coordinate axes at \(\mathrm{A}\) and \(\mathrm{B}\), then the locus of the foot of perpendicular from \(\mathrm{O}\) on \(\mathrm{AB}\) is : (a) \(\left(x^{2}+y^{2}\right)^{2}=4 R^{2} x^{2} y^{2}\) (b) \(\left(x^{2}+y^{2}\right)^{3}=4 R^{2} x^{2} y^{2}\) (c) \(\left(x^{2}+y^{2}\right)^{2}=4 R x^{2} y^{2}\) (d) \(\left(x^{2}+y^{2}\right)(x+y)=R^{2} x y\)

Let \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) be the centres of the circles \(\mathrm{x}^{2}+y^{2}-2 \mathrm{x}-2 \mathrm{y}-2=0\) and \(x^{2}+y^{2}-6 x-6 y+14=0\) respectively. If \(P\) and \(Q\) are the points of intersection of these circles then, the area (in sq. units) of the quadrilateral \(\mathrm{PC}_{1} \mathrm{QC}_{2}\) is: (a) 8 (b) 6 (c) 9 (d) 4

If a variable line, \(3 x+4 y-\lambda=0\) is such that the two circles \(x^{2}+y^{2}-2 x-2 y+1=0\) and \(x^{2}+y^{2}-18 x-2 y+78=0\) are on its opposite sides, then the set of all values of \(\lambda\) is the interval : \(\quad\) [Jan. 12, 2019 (I)] (a) \((2,17)\) (b) \([13,23]\) (c) \([12,21]\) (d) \((23,31)\)

A square is inscribed in the circle \(x^{2}+y^{2}-6 x+8 y-103=0\) with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is : (a) 6 (b) \(\sqrt{137}\) (c) \(\sqrt{41}\) (d) 13

Two circles with equal radii are intersecting at the points \((0,1)\) and \((0,-1)\). The tangent at the point \((0,1)\) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is : (a) 1 (b) 2 (c) \(2 \sqrt{2}\) (d) \(\sqrt{2}\)

A circle cuts a chord of length \(4 \mathrm{a}\) on the \(x\)-axis and passes through a point on the \(y\)-axis, distant \(2 \mathrm{~b}\) from the origin. Then the locus of the centre of this circle, is : (a) a hyperbola (b) an ellipse (c) a straight line (d) a parabola

If a circle C passing through the point \((4,0)\) touches the circle \(x^{2}+y^{2}+4 x-6 y=12\) externally at the point \((1,-1)\), then the radius of \(\mathrm{C}\) is: (a) \(2 \sqrt{5}\) (b) 4 (c) 5 (d) \(\sqrt{57}\)

Three circles of radii \(\mathrm{a}, \mathrm{b}, \mathrm{c}(\mathrm{a}<\mathrm{b}<\mathrm{c})\) touch each other externally. If they have \(x\)-axis as a common tangent, then: (a) \(\frac{1}{\sqrt{\mathrm{a}}}=\frac{1}{\sqrt{\mathrm{b}}}+\frac{1}{\sqrt{\mathrm{c}}}\) (b) \(\frac{1}{\sqrt{b}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{c}}\) (c) \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in A.P (d) \(\sqrt{a}, \sqrt{b}, \sqrt{c}\) are in A.P.

If the circles \(x^{2}+y^{2}-16 x-20 y+164=r^{2}\) and \((x-4)^{2}+(y-7)^{2}=36\) intersect at two distinct points, then: (a) \(\mathrm{r}>11\) (b) \(0<\mathrm{r}<1\) (c) \(\mathrm{r}=11\) (d) \(1

The straight line \(x+2 y=1\) meets the coordinate axes at \(A\) and \(\mathrm{B}\). A circle is drawn through \(\mathrm{A}, \mathrm{B}\) and the origin. Then the sum of perpendicular distances from \(\mathrm{A}\) and \(\mathrm{B}\) on the tangent to the circle at the origin is : (a) \(\frac{\sqrt{5}}{2}\) (b) \(2 \sqrt{5}\) (c) \(\frac{\sqrt{5}}{4}\) (d) \(4 \sqrt{5}\)

If the tangent at \((1,7)\) to the curve \(x^{2}=y-6\) touches the circle \(x^{2}+y^{2}+16 x+12 y+c=0\) then the value of \(c\) is : (a) 185 (b) 85 (c) 95 (d) 195

If a circle \(\mathrm{C}\), whose radius is 3 , touches externally the circle, \(x^{2}+y^{2}+2 x-4 y-4=0\) at the point \((2,2)\), then the length of the intercept cut by this circle \(c\), on the \(x\)-axis is equal to (a) \(\sqrt{5}\) (b) \(2 \sqrt{3}\) (c) \(3 \sqrt{2}\) (d) \(2 \sqrt{5}\)

A circle passes through the points \((2,3)\) and \((4,5) .\) If its centre lies on the line, \(y-4 x+3=0\), then its radius is equal to (a) \(\sqrt{5}\) (b) 1 (c) \(\sqrt{2}\) (d) 2

The tangent to the circle \(C_{1}: x^{2}+y^{2}-2 x-1=0\) at the point (2, 1) cuts off a chord of length 4 from a circle \(C_{2}\) whose centre is \((3,-2)\). The radius of \(C_{2}\) is (a) \(\sqrt{6}\) (b) 2 (c) \(\sqrt{2}\) (d) 3

The radius of a circle, having minimum area, which touches the curve \(y=4-x^{2}\) and the lines, \(y=|x|\) is: (a) \(4(\sqrt{2}+1)\) (b) \(2(\sqrt{2}+1)\) (c) \(2(\sqrt{2}-1)\) (d) \(4(\sqrt{2}-1)\)

The equation \(\operatorname{Im}\left(\frac{i z-2}{z-i}\right)+1=0, z \in C, z \neq i\) represents a part of a circle having radius equal to : (a) 2 (b) 1 (c) \(\frac{3}{4}\) (d) \(\frac{1}{2}\)

A line drawn through the point \(\mathrm{P}(4,7)\) cuts the circle \(x^{2}+y^{2}=9\) at the points \(A\) and \(B\). Then \(P A \cdot P B\) is equal to : (a) 53 (b) 56 (c) 74 (d) 65

Let \(\mathrm{z} \in \mathrm{C}\), the set of complex numbers. Then the equation, \(2|z+3 i|-|z-i|=0\) represents: (a) a circle with radius \(\frac{8}{3}\). (b) a circle with diameter \(\frac{10}{3}\). (c) an ellipse with length of major axis \(\frac{16}{3}\). (d) an ellipse with length of minor axis \(\frac{16}{9}\)

If a point \(\mathrm{P}\) has co-ordinates \((0,-2)\) and \(\mathrm{Q}\) is any point on the circle, \(x^{2}+y^{2}-5 x-y+5=0\), then the maximum value \(\begin{aligned}&\text { of }(\mathrm{PQ})^{2} \text { is : (a) \)\frac{25+\sqrt{6}}{2}\( (b) \)14+5 \sqrt{3}\( (c) \)\frac{47+10 \sqrt{6}}{2}\( (d) \)8+5 \sqrt{3}$

If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles \(\cos ^{-1}\left(\frac{1}{7}\right)\) and \(\sec ^{-1}(7)\) at the centre respectively, then the distance between these chords, is: (a) \(\frac{4}{\sqrt{7}}\) (b) \(\frac{8}{\sqrt{7}}\) (c) \(\frac{8}{7}\) (d) \(\frac{16}{7}\)

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) 5 (b) 10 (c) \(5 \sqrt{2}\) (d) \(5 \sqrt{3}\)

A circle passes through \((-2,4)\) and touches the \(y\)-axis at \((0,2)\). Which one of the following equations can represent a diameter of this circle? (a) \(2 x-3 y+10=0\) (b) \(3 x+4 y-3=0\) (c) \(4 x+5 y-6=0\) (d) \(5 x+2 y+4=0\)

If the incentre of an equilateral triangle is \((1,1)\) and the equation of its one side is \(3 x+4 y+3=0\), then the equation of the circumcircle of this triangle is: (a) \(x^{2}+y^{2}-2 x-2 y-14=0\) (b) \(x^{2}+y^{2}-2 x-2 y-2=0\) (c) \(x^{2}+y^{2}-2 x-2 y+2=0\) (d) \(x^{2}+y^{2}-2 x-2 y-7=0\)

If a circle passing through the point \((-1,0)\) touches \(\mathrm{y}\) axis at \((0,2)\), then the length of the chord of the circle along the x-axis is : (a) \(\frac{3}{2}\) (b) 3 (c) \(\frac{5}{2}\) (d) 5

Let the tangents drawn to the circle, \(\mathrm{x}^{2}+\mathrm{y}^{2}=16\) from the point \(\mathrm{P}(0, \mathrm{~h})\) meet the \(\mathrm{x}\)-axis at point \(\mathrm{A}\) and \(\mathrm{B}\). If the area of \(\triangle \mathrm{APB}\) is minimum, then \(\mathrm{h}\) is equal to : (a) \(4 \sqrt{2}\) (b) \(3 \sqrt{3}\) (c) \(3 \sqrt{2}\) (d) \(4 \sqrt{3}\)

If \(y+3 x=0\) is the equation of a chord of the circle, \(x^{2}+y^{2}-30 x=0\), then the equation of the circle with this chord as diameter is: (a) \(x^{2}+y^{2}+3 x+9 y=0\) (b) \(x^{2}+y^{2}+3 x-9 y=0\) (c) \(x^{2}+y^{2}-3 x-9 y=0\) (d) \(x^{2}+y^{2}-3 x+9 y=0\)

The largest value of for which the region represented by the set \(\\{\omega \in C|\omega-4-i| \leq r\\}\) is contained in the region represented by the set \((z \in c /|z-1| \leq|z+i|)\), is equal to: (a) \(\frac{5}{2} \sqrt{2}\) (b) \(2 \sqrt{2}\) (c) \(\frac{3}{2} \sqrt{2}\) (d) \(\sqrt{17}\)

Let \(C\) be the circle with centre at \((1,1)\) and radius \(=1\). If \(T\) is the circle centred at \((0, y)\), passing through origin and touching the circle \(C\) externally, then the radius of \(T\) is equal to (a) \(\frac{1}{2}\) (b) \(\frac{1}{4}\) (c) \(\frac{\sqrt{3}}{\sqrt{2}}\) (d) \(\frac{\sqrt{3}}{2}\)

The equation of circle described on the chord \(3 x+y+5=0\) of the circle \(x^{2}+y^{2}=16\) as diameter is: (a) \(x^{2}+y^{2}+3 x+y-11=0\) (b) \(x^{2}+y^{2}+3 x+y+1=0\) (c) \(x^{2}+y^{2}+3 x+y-2=0\) (d) \(x^{2}+y^{2}+3 x+y-22=0\)

For the two circles \(x^{2}+y^{2}=16\) and \(\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{y}=0\), there is/are (a) one pair of common tangents (b) two pair of common tangents (c) three pair of common tangents (d) no common tangent

The set of all real values of \(\lambda\) for which exactly two common tangents can be drawn to the circles \(x^{2}+y^{2}-4 x-4 y+6=0\) and \(x^{2}+y^{2}-10 x-10 y+\lambda=0\) is the interval: (a) \((12,32)\) (b) \((18,42)\) (c) \((12,24)\) (d) \((18,48)\)

If the point \((1,4)\) lies inside the circle \(\mathrm{x}^{2}+\mathrm{y}^{2}-6 \mathrm{x}-10 \mathrm{y}+\mathrm{P}=0\) and the circle does not touch or intersect the coordinate axes, then the set of all possible values of P is the interval: (a) \((0,25)\) (b) \((25,39)\) (c) \((9,25)\) (d) \((25,29)\)

Let a and \(\mathrm{b}\) be any two numbers satisfying \(\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}=\frac{1}{4}\). Then, the foot of perpendicular from the origin on the variable line, \(\frac{x}{a}+\frac{y}{b}=1\), lies on: (a) a hyperbola with each semi-axis \(=\sqrt{2}\) (b) a hyperbola with each semi-axis \(=2\) (c) a circle of radius \(=2\) (d) a circle of radius \(=\sqrt{2}\)

The circle passing through \((1,-2)\) and touching the axis of \(x\) at \((3,0)\) also passes through the point (a) \((-5,2)\) (b) \((2,-5)\) (c) \((5,-2)\) (d) \((-2,5)\)

If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle \(60^{\circ}\) on the circumference of the first circle, then the radius of the arc is: (a) \(\sqrt{3}\) (b) \(\frac{1}{2}\) (c) 1 (d) \(\sqrt{2}\)

Statement 1: The only circle having radius \(\sqrt{10}\) and a diameter along line \(2 x+y=5\) is \(x^{2}+y^{2}-6 x+2 y=0\). Statement \(\mathbf{2}: 2 x+y=5\) is a normal to the circle $x^{2}+y^{2}-6 x+2 y=0 . (a) Statement 1 is false; Statement 2 is true. (b) Statement 1 is true; Statement 2 is true, Statement 2 is a correct explanation for Statement 1 . (c) Statement 1 is true; Statement 2 is false. (d) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 .