Chapter 11

If the circle \(x^{2}+y^{2}-6 x-8 y+\left(25-a^{2}\right)=0\) touches the axis of \(x\), then a equals. (a) 0 (b) \(\pm 4\) (c) \(\pm 2\) (d) \(\pm 3\)

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

If two vertices of an equilateral triangle are \(\mathrm{A}(-a, 0)\) and \(B(a, 0), a>0\), and the third vertex \(\mathrm{C}\) lies above \(x\)-axis then the equation of the circumcircle of \(\triangle \mathrm{ABC}\) is: (a) \(3 x^{2}+3 y^{2}-2 \sqrt{3} a y=3 a^{2}\) (b) \(3 x^{2}+3 y^{2}-2 a y=3 a^{2}\) (c) \(x^{2}+y^{2}-2 a y=a^{2}\) (d) \(x^{2}+y^{2}-\sqrt{3} a y=a^{2}\)

If each of the lines \(5 x+8 y=13\) and \(4 x-y=3\) contains a diameter of the circle \(x^{2}+y^{2}-2\left(a^{2}-7 a+11\right) x-2\left(a^{2}-6 a+6\right) y+b^{3}+1=0\), then (a) \(a=5\) and \(b \notin(-1,1)\) (b) \(a=1\) and \(b \notin(-1,1)\) (c) \(a=2\) and \(b \notin(-\infty, 1)\) (d) \(a=5\) and \(b \in(-\infty, 1)\)

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: (a) \(\frac{10}{3}\) (b) \(\frac{3}{5}\) (c) \(\frac{6}{5}\) (d) \(\frac{5}{3}\)

The number of common tangents of the circles given by \(x^{2}+y^{2}-8 x-2 y+1=0\) and \(x^{2}+y^{2}+6 x+8 y=0\) is (a) one (b) four (c) two (d) three

If the line \(y=m x+1\) meets the circle \(x^{2}+y^{2}+3 x=0\) in two points equidistant from and on opposite sides of \(x\)-axis, then (a) \(3 m+2=0\) (b) \(3 m-2=0\) (c) \(2 m+3=0\) (d) \(2 m-3=0\)

If three distinct points \(A, B, C\) are given in the 2 -dimensional coordinate plane such that the ratio of the distance of each one of them from the point \((1,0)\) to the distance from \((-1,0)\) is equal to \(\frac{1}{2}\), then the circumcentre of the triangle \(A B C\) is at the point (a) \(\left(\frac{5}{3}, 0\right)\) (b) \((0,0)\) (c) \(\left(\frac{1}{3}, 0\right)\) (d) \((3,0)\)

The equation of the circle passing through the point \((1,2)\) and through the points of intersection of \(x^{2}+y^{2}-4 x-6 y-21=0\) and \(3 x+4 y+5=0\) is given by (a) \(x^{2}+y^{2}+2 x+2 y+11=0\) (b) \(x^{2}+y^{2}-2 x+2 y-7=0\) (c) \(x^{2}+y^{2}+2 x-2 y-3=0\) (d) \(x^{2}+y^{2}+2 x+2 y-11=0\)

The equation of the circle passing through the point \((1,0)\) and \((0,1)\) and having the smallest radius is - (a) \(x^{2}+y^{2}-2 x-2 y+1=0\) (b) \(x^{2}+y^{2}-x-y=0\) (c) \(x^{2}+y^{2}+2 x+2 y-7=0\) (d) \(x^{2}+y^{2}+x+y-2=0\)

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 circle \(x^{2}+y^{2}=4 x+8 y+5\) intersects the line \(3 x-4 y=m\) at two distinct points if (a) \(-35

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 \(\mathrm{P}, \mathrm{Q}\) and \((1,1)\) for: (a) all except one value of \(p\) (b) all except two values of \(p\) (c) exactly one value of \(p\) (d) all values of \(p\)

Three distinct points \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are given in the 2-dimensional coordinates plane such that the ratio of the distance of any one of them from the point \((1,0)\) to the distance from the point \((-1,0)\) is equal to \(\frac{1}{3}\). Then the circumcentre of the triangle \(\mathrm{ABC}\) is at the point: (a) \(\left(\frac{5}{4}, 0\right)\) (b) \(\left(\frac{5}{2}, 0\right)\) (c) \(\left(\frac{5}{3}, 0\right)\) (d) \((0,0)\)

The point diametrically opposite to the point \(P(1,0)\) on the circle \(x^{2}+y^{2}+2 x+4 y-3=0\) is (a) \((3,-4)\) (b) \((-3,4)\) (c) \((-3,-4)\) (d) \((3,4)\)

Consider a family of circles which are passing through the point \((-1,1)\) and are tangent to \(x\)-axis. If \((h, k)\) are the coordinate of the centre of the circles, then the set of values of \(k\) is given by the interval (a) \(-\frac{1}{2} \leq k \leq \frac{1}{2}\) (b) \(k \leq \frac{1}{2}\) (c) \(0 \leq k \leq \frac{1}{3}\) (d) \(k \geq \frac{1}{2}\)

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 center 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}$

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 (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) \(x^{2}+y^{2}-2 x+2 y-47=0\)

If the pair of lines \(a x^{2}+2(a+b) x y+b y^{2}=0\) lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then (a) \(3 a^{2}-10 a b+3 b^{2}=0\) (b) \(3 a^{2}-2 a b+3 b^{2}=0\) (c) \(3 a^{2}+10 a b+3 b^{2}=0\) (d) \(3 a^{2}+2 a b+3 b^{2}=0\)

If the circles \(x^{2}+y^{2}+2 a x+c y+a=0\) and \(x^{2}+y^{2}-3 a x+d y-1=0\) intersect in two distinct points \(P\) and \(Q\) then the line \(5 x+b y-a=0\) passes through \(P\) and \(Q\) for (a) exactly one value of \(a\) (b) no value of \(a\) (c) infinitely many values of \(a\) (d) exactly two values of \(a\)

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) \((y-q)^{2}=4 p x\) (b) \((x-q)^{2}=4 p y\) (c) \((y-p)^{2}=4 q x\) (d) \((x-p)^{2}=4 q y\)

If the lines \(2 x+3 y+1=0\) and \(3 x-y-4=0\) lie along diameter of a circle of circumference \(10 \pi\), then the equation of the circle is (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) \(x^{2}+y^{2}-2 x+2 y-23=0\)

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 (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) \(x^{2}+y^{2}-x-y=0\)

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 point, then (a) \(r>2\) (b) \(2

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) \(x^{2}+y^{2}+2 x-2 y=62\) (c) \(x^{2}+y^{2}+2 x-2 y=47\) (d) \(x^{2}+y^{2}-2 x+2 y=47\).

If the chord \(y=m x+1\) of the circle \(x^{2}+y^{2}=1\) subtends an angle of measure \(45^{\circ}\) at the major segment of the circle then value of \(m\) is (a) \(2 \pm \sqrt{2}\) (b) \(-2 \pm \sqrt{2}\) (c) \(-1 \pm \sqrt{2}\) (d) none of these

The centres of a set of circles, each of radius 3 , lie on the circle \(x^{2}+y^{2}=25\). The locus of any point in the set is (a) \(4 \leq x^{2}+y^{2} \leq 64\) (b) \(x^{2}+y^{2} \leq 25\) (c) \(x^{2}+y^{2} \geq 25\) (d) \(3 \leq x^{2}+y^{2} \leq 9\)

The centre of the circle passing through \((0,0)\) and \((1,0)\) and touching the circle \(x^{2}+y^{2}=9\) is (a) \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (b) \(\left(\frac{1}{2},-\sqrt{2}\right)\) (c) \(\left(\frac{3}{2}, \frac{1}{2}\right)\) (d) \(\left(\frac{1}{2}, \frac{3}{2}\right)\)

The equation of a circle with origin as a centre and passing through equilateral triangle whose median is of length \(3 a\) is (a) \(x^{2}+y^{2}=9 a^{2}\) (b) \(x^{2}+y^{2}=16 a^{2}\) (c) \(x^{2}+y^{2}=4 a^{2}\) (d) \(x^{2}+y^{2}=a^{2}\)

Let \(L_{1}\) be a tangent to the parabola \(y^{2}=4(x+1)\) and \(L_{2}\) be a tangent to the parabola \(y^{2}=8(x+2)\) such that \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) intersect at right angles. Then \(L_{1}\) and \(L_{2}\) meet on the straight line : (a) \(x+3=0\) (b) \(2 x+1=0\) (c) \(x+2=0\) (d) \(x+2 y=0\)

The centre of the circle passing through the point \((0,1)\) and touching the parabola \(y=x^{2}\) at the point \((2,4)\) is: (a) \(\left(\frac{-53}{10}, \frac{16}{5}\right)\) (b) \(\left(\frac{6}{5}, \frac{53}{10}\right)\) (c) \(\left(\frac{3}{10}, \frac{16}{5}\right)\) (d) \(\left(\frac{-16}{5}, \frac{53}{10}\right)\)

If the common tangent to the parabolas, \(y^{2}=4 x\) and \(x^{2}=4 y\) also touches the circle, \(x^{2}+y^{2}=\mathrm{c}^{2}\), then \(\mathrm{c}\) is equal to: (a) \(\frac{1}{2 \sqrt{2}}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{2}\)

Let \(P\) be a point on the parabola, \(y^{2}=12 x\) and \(N\) be the foot of the perpendicular drawn from \(P\) on the axis of the parabola. A line is now drawn through the mid-point \(M\) of \(P N\), parallel to its axis which meets the parabola at \(Q\). If the \(y\)-intercept of the line \(N Q\) is \(\frac{4}{3}\), then : (a) \(P N=4\) (b) \(M Q=\frac{1}{3}\) (c) \(M Q=\frac{1}{4}\) (d) \(P N=3\)

The area (in sq. units) of an equilateral triangle inscribed in the parabola \(y^{2}=8 x\), with one of its vertices on the vertex of this parabola, is: (a) \(64 \sqrt{3}\) (b) \(256 \sqrt{3}\) (c) \(192 \sqrt{3}\) (d) \(128 \sqrt{3}\)

If one end of a focal chord \(\mathrm{AB}\) of the parabola \(y^{2}=8 x\) is at \(\mathrm{A}\left(\frac{1}{\sqrt{2}},-2\right)\), then the equation of the tangent to it at \(\mathrm{B}\) is: (a) \(2 x+y-24=0\) (b) \(x-2 y+8=0\) (c) \(x+2 y+8=0\) (d) \(2 x-y-24=0\)

The locus of a point which divides the line segment joining the point \((0,-1)\) and a point on the parabola, \(x^{2}=4 y\), internally in the ratio \(1: 2\), is: (a) \(9 x^{2}-12 y=8\) (b) \(9 x^{2}-3 y=2\) (c) \(x^{2}-3 y=2\) (d) \(4 x^{2}-3 y=2\)

Let a line \(y=m x(m>0)\) intersect the parabola, \(y^{2}=x\) at a point \(P\), other than the origin. Let the tangent to it at \(P\) meet the \(x\)-axis at the point \(Q\), If area \((\Delta O P Q)=4 \mathrm{sq}\). units, then \(m\) is equal to ___ .

If \(y=m x+4\) is a tangent to both the parabolas, \(y^{2}=4 x\) and \(x^{2}=2 b y\), then \(b\) is equal to: (a) \(-32\) (b) \(-64\) (c) \(-128\) (d) 128

The tangents to the curve \(y=(x-2)^{2}-1\) at its points of intersection with the line \(x-y=3\), intersect at the point : (b) \(\left(-\frac{5}{2},-1\right)\) (a) \(\left(\frac{5}{2}, 1\right)\) (c) \(\left(\frac{5}{2},-1\right)\) (d) \(\left(-\frac{5}{2}, 1\right)\)

If the line \(a x+y=c\), touches both the curves \(x^{2}+y^{2}=1\) and \(y^{2}=4 \sqrt{2} x\), then \(|c|\) is equal to (a) 2 (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{2}\) (d) \(\sqrt{2}\)

The area (in sq. units) of the smaller of the two circles that touch the parabola, \(y^{2}=4 x\) at the point \((1,2)\) and the \(x\)-axis is: (a) \(8 \pi(2-\sqrt{2})\) (b) \(4 \pi(2-\sqrt{2})\) (c) \(4 \pi(3+\sqrt{2})\) (d) \(8 \pi(3-2 \sqrt{2})\)

If one end of a focal chord of the parabola, \(y^{2}=16 x\) is at \((1,4)\), then the length of this focal chord is: (a) 25 (b) 22 (c) 24 (d) 20

The equation of a tangent to the parabola, \(x^{2}=8 y\), which makes an angle \(\theta\) with the positive direction of \(x\)-axis, is: (a) \(y=x \tan \theta+2 \cot \theta\) (b) \(y=x \tan \theta-2 \cot \theta\) (c) \(x=y \cot \theta+2 \tan \theta\) (d) \(x=y \cot \theta-2 \tan \theta\)

Equation of a common tangent to the parabola \(y^{2}=4 x\) and the hyperbola \(x y=2\) is : (a) \(x+y+1=0\) (b) \(x-2 y+4=0\) (c) \(x+2 y+4=0\) (d) \(4 x+2 y+1=0\)

If the area of the triangle whose one vertex is at the vertex of the parabola, \(y^{2}+4\left(x-a^{2}\right)=0\) and the other two vertices are the points of intersection of the parabola and \(y\)-axis, is 250 sq. units, then a value of ' \(a\) ' is : (a) \(5 \sqrt{5}\) (b) \(5\left(2^{1-3}\right)\) (c) \((10)^{23}\) (d) 5

If the parabolas \(y^{2}=4 b(x-c)\) and \(y^{2}=8 a x\) have a common normal, then which one of the following is a valid choice for the ordered triad \((a, b, c) ? (a) \)\left(\frac{1}{2}, 2,3\right)\( (b) \)(1,1,3)\( (c) \)\left(\frac{1}{2}, 2,0\right)\( (d) \)(1,1,0)$

The length of the chord of the parabola \(x^{2}=4 y\) having equation \(x-\sqrt{2} y+4 \sqrt{2}=0\) is: (a) \(3 \sqrt{2}\) (b) \(2 \sqrt{11}\) (c) \(8 \sqrt{2}\) (d) \(6 \sqrt{3}\)

Axis of a parabola lies along \(x\)-axis. If its vertex and focus are at distance 2 and 4 respectively from the origin, on the positive \(x\)-axis then which of the following points does not lie on it? (a) \((5,2 \sqrt{6})\) (b) \((8,6)\) (c) \((6,4 \sqrt{2})\) (d) \((4,-4)\)

Equation of a common tangent to the circle, \(x^{2}+y^{2}-6 x=0\) and the parabola, \(y^{2}=4 x\), is : (a) \(2 \sqrt{3} y=12 x+1\) (b) \(\sqrt{3} y=x+3\) (c) \(2 \sqrt{3} y=-x-12\) (d) \(\sqrt{3} y=3 x+1\)