Chapter 19

An isosceles \(\triangle A B C\) is inscribed in a circle \(x^{2}+y^{2}=a^{2}\) with the vertex \(A\) at \((a, 0)\) and the base angles \(B\) and \(C\) each equal to \(75^{\circ}\) then length of the base \(B C\) is (A) \(\frac{a}{2}\) (B) \(a\) (C) \(\frac{2 a}{\sqrt{3}}\) (D) \(\frac{\sqrt{3} a}{2}\)

Let \(a_{n}, n=1,2,3,4\) represent four distinct positive real numbers other than unity such that each pair of the logarithm of \(a_{n}\) and the reciprocal of logarithm denotes a point on a circle, whose centre lies on \(y\)-axis. The product of these four numbers is (A) 0 (B) 1 (C) 2 (D) 13

If the tangents \(P A\) and \(P B\) are drawn from the point \(P(-1,2)\) to the circle \(x^{2}+y^{2}+x-2 y-3=0\) and \(C\) is the centre of the circle, then the area of the quadrilateral \(P A C B\) is (A) 4 (B) 16 (C) does not exist (D) none of these

If the line \((y-2)=m(x+1)\) intersects the circle \(x^{2}+\) \(y^{2}+2 x-4 y-3=0\) at two real distinct points, then the number of possible values of \(m\) is (A) 2 (B) 1 (C) any real value of \(m\) (D) none of these

The number of points on the circle \(x^{2}+y^{2}-4 x-10 y+\) \(13=0\) which are at a distance 1 from the point \((-3,2)\) is (A) \(\underline{1}\) (B) 2 (C) 3 (D) none of these

If the equations of four circles are \((x \pm 4)^{2}+(y \pm 4)^{2}\) \(=4^{2}\), then the radius of the smallest circle touching all the four circles is (A) \(4(\sqrt{2}+1)\) (B) \(4(\sqrt{2}-1)\) (C) \(2(\sqrt{2}-1)\) (D) none of these

The intercept on the line \(y=x\) by the circle \(x^{2}+y^{2}-2 x=\) 0 is \(A B\). Equation of the circle with \(A B\) 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) none of these

The locus of the mid-point of the chord of the circle \(x^{2}\) \(+y^{2}-2 x-2 y-2=0\) which makes an angle of \(120^{\circ}\) at the centre is (A) \(x^{2}+y^{2}-2 x-2 y+1=0\) (B) \(x^{2}+y^{2}+x+y-1=0\) (C) \(x^{2}+y^{2}-2 x-2 y-1=0\) (D) none of these

A square is inscribed in the circle \(x^{2}+y^{2}-2 x+4 y+3\) \(=0 .\) 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

If the lines \(a_{1} x+b_{1} y+c_{1}=0\) and \(a_{2} x+b_{2} y+c_{2}=0\) cut the coordinate axes in concyclic points, then (A) \(a_{1} a_{2}=b_{1} b_{2}\) (B) \(a_{1} b_{1}=a_{2} b_{2}\) (C) \(a_{1} b_{2}=a_{2} b_{1}\) (D) none of these

The circle \(x^{2}+y^{2}=4\) cuts the line joining the points \(A(1,0)\) and \(B(3,4)\) in two points \(P\) and \(Q\). Let \(\frac{B P}{P A}=\alpha\) and \(\frac{B Q}{Q A}=\beta .\) Then, \(\alpha\) and \(\beta\) are roots of the quadratic equation (A) \(3 x^{2}+2 x-21=0\) (B) \(3 x^{2}+2 x+21=0\) (C) \(2 x^{2}+3 x-21=0\) (D) none of these

If the equation of the incircle of an equilateral triangle is \(x^{2}+y^{2}+4 x-6 y+4=0\), then the equation of the circumcircle of the triangle is (A) \(x^{2}+y^{2}+4 x+6 y-23=0\) (B) \(x^{2}+y^{2}+4 x-6 y-23=0\) (C) \(x^{2}+y^{2}-4 x-6 y-23=0\) (D) none of these

Two distinet chords drawn from the point \((p, q)\) on the circle \(x^{2}+y^{2}=p x+q y\), where \(p q \neq 0\), are bisected by the \(x\)-axis. Then, (A) \(|p|=|q|\) (B) \(p^{2}=8 q^{2}\) (C) \(p^{2}<8 q^{2}\) (D) \(p^{2}>8 q^{2}\)

For the two circles \(x^{2}+y^{2}=16\) and \(x^{2}+y^{2}-2 y=0\) there is/are (A) one pair of common tangents (B) two pairs of common tangents (C) three common tangents (D) no common tangent

Let \(A B\) be a chord of the circle \(x^{2}+y^{2}=r^{2}\) subtending a right angle at the centre. Then, the locus of the centroid of the \(\Delta P A B\) as \(P\) moves on the circle is (A) a parabola (B) a circle (C) an ellipse (D) a pair of straight lines

The equation of the smallest circle passing through the intersection of the line \(x+y=1\) and the circle \(x^{2}+y^{2}=\) 9 is (A) \(x^{2}+y^{2}+x+y-8=0\) (B) \(x^{2}+y^{2}-x-y-8=0\) (C) \(x^{2}+y^{2}-x-y+8=0\) (D) none of these

If \((a, b)\) is a point on the circle whose centre is on the \(x\)-axis and which touches the line \(x+y=0\) at \((2,-2)\), then the greatest value of \(a\) is (A) \(4+2 \sqrt{2}\) (B) \(2+2 \sqrt{2}\) (C) \(4+\sqrt{2}\) (D) none of these

If \(a>2 b>0\) then the positive value of \(m\) for which \(y=m x-b \sqrt{1+m^{2}}\) is a common tangent to \(x^{2}+y^{2}=\) \(b^{2}\) and \((x-a)^{2}+y^{2}=b^{2}\) is (A) \(\frac{2 b}{\sqrt{a^{2}-4 b^{2}}}\) (B) \(\frac{\sqrt{a^{2}-4 b^{2}}}{2 b}\) (C) \(\frac{2 b}{a-2 b}\) (D) \(\frac{b}{a-2 b}\)

If the locus of a point which moves so that the line joining the points of contact of the tangents drawn from it to the circle \(x^{2}+y^{2}=b^{2}\) touches the circle \(x^{2}+\) \(y^{2}=a^{2}\), is the circle \(x^{2}+y^{2}=c^{2}\), then \(a, b, c\) are in (A) A. P. (B) G. P. (C) H. P. (D) none of these

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

The point \((1,4)\) lies inside the circle \(x^{2}+y^{2}-6 x-10 y\) \(+p=0\) which does not touch or intersect the coordinate axes, then (A) \(0

The equation to the sides \(A B, B C, C A\) of a \(\triangle A B C\) are \(x+y=1,4 x-y+4=0\) and \(2 x+3 y=6\). Circles are drawn on \(A B, B C, C A\) as diameters. The point of concurrence of the common chords is (A) centroid of the triangle (B) orthocentre (C) circumcentre (D) incentre

The coordinates of the point on the circle \(x^{2}+y^{2}-2 x\) \(-4 y-11=0\) farthest from the origin are (A) \(\left(2+\frac{8}{\sqrt{5}}, 1+\frac{4}{\sqrt{5}}\right)\) (B) \(\left(1+\frac{4}{\sqrt{5}}, 2+\frac{8}{\sqrt{5}}\right)\) (C) \(\left(1+\frac{8}{\sqrt{5}}, 2+\frac{4}{\sqrt{5}}\right)\) (D) none of these

If the line \(3 x+a y-20=0\) cuts the circle \(x^{2}+y^{2}=25\) at real, distinct or coincident points, then \(a\) belongs to the interval (A) \([-\sqrt{7}, \sqrt{7}]\) (B) \((-\sqrt{7}, \sqrt{7})\) (C) \((-\infty-\sqrt{7}] \cup[\sqrt{7}, \infty)\) (D) none of these

The locus of centre of the circle which touches the circle \(x^{2}+(y-1)^{2}=1\) externally and also touches \(x\)-axis is (A) \(\left\\{(x, y): x^{2}+(y-1)^{2}=4\right\\} \cup\\{(x, y): y<0\\}\) (B) \(\left\\{(x, y): x^{2}=4 y\right\\} \cup\\{(0, y): y<0\\}\) (C) \(\left\\{(x, y): x^{2}=y\right\\} \cup\\{(0, y): y<0\\}\) (D) \(\left\\{(x, y): x^{2}=4 y\right\\} \cup\\{(x, y) ; y<0\\}\)

Let \(P Q\) and \(R S\) be tangents at the extremeties of the diameter \(P R\) of a circle of radius \(r\). If \(P S\) and \(R Q\) intersect at a point \(X\) on the circumference of the circle, then \(2 r\) equals (A) \(\sqrt{P Q \cdot R S}\) (B) \(\frac{P Q+R S}{2}\) (C) \(\frac{2 P Q \cdot R S}{P Q+R S}\) (D) \(\sqrt{\frac{P Q^{2}+R S^{2}}{2}}\)

Circles are drawn through the point \((-5,0)\) to cut the \(x\)-axis on the positive side and making an intercept of 10 units on the \(x\)-axis. The equation of the locus of the centre of these circles is (A) \(x+y=0\) (B) \(x-y=0\) (C) \(x=0\) (D) \(y=0\)

The circle \(x^{2}+y^{2}-4 x-8 y+16=0\) rolls up the tangent to it at \((2+\sqrt{3}, 3)\) by 2 units, assuming the \(x\)-axis as horizontal, the equation of the circle in the new position is (A) \(x^{2}+y^{2}-6 x-2(4+\sqrt{3}) y+24+8 \sqrt{3}=0\) (B) \(x^{2}+y^{2}+6 x-2(4+\sqrt{3}) y+24+8 \sqrt{3}=0\) (C) \(x^{2}+y^{2}-6 x+2(4+\sqrt{3}) y+24+8 \sqrt{3}=0\) (D) none of these

The equation of the circle, passing through the point \((2,8)\), touching the lines \(4 x-3 y-24=0\) and \(4 x+3 y\) \(-42=0\) and having \(x\) coordinate of the centre of the circle numerically less then or equal to 8 , is (A) \(x^{2}+y^{2}+4 x-6 y-12=0\) (B) \(x^{2}+y^{2}-4 x+6 y-12=0\) (C) \(x^{2}+y^{2}-4 x-6 y-12=0\) (D) none of these

If the line \(\frac{x}{a}+\frac{y}{b}=1\) moves in such a way that \(\frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{1}{c^{2}}\), where \(c\) is a constant, then the locus of the foot of the perpendicular from the origin on the straight line describes the circle (A) \(x^{2}+y^{2}=4 c^{2}\) (B) \(x^{2}+y^{2}=2 c^{2}\) (C) \(x^{2}+y^{2}=c^{2}\) (D) none of these

A circle touches both the \(x\)-axis and the line \(4 x-3 y+\) \(4=0 .\) If its centre is in the third quadrant and lies on the line \(x-y-1=0\), then the equation of the circle is (A) \(9\left(x^{2}+y^{2}\right)+6 x+24 y-1=0\) (B) \(9\left(x^{2}+y^{2}\right)+6 x-24 y+1=0\) (C) \(9\left(x^{2}+y^{2}\right)+6 x+24 y+1=0\) (D) none of these

If \(\theta_{1}, \theta_{2}\) be the inclinations of tangents drawn from the point \(\vec{P}\) to the circle \(x^{2}+y^{2}=a^{2}\) and \(\cot \theta_{1}+\cot \theta_{2}=k\), then the locus of \(P\) is (A) \(k\left(y^{2}+a^{2}\right)=2 x y\) (B) \(k\left(y^{2}-a^{2}\right)=2 x y\) (C) \(k\left(y^{2}-a^{2}\right)=x y\) (D) none of these

If the chord of contact of tangents from a point on the circle \(x^{2}+y^{2}=a^{2}\) to the circle \(x^{2}+y^{2}=b^{2}\) touches the circle \(x^{2}+y^{2}=c^{2}\), then \(a, b, c\) are in (A) A. \(P\). (B) G. P. (C) H. P. (D) none of these

To which of the following circles, the line \(y-x+3=\) 0 is normal at the point \(\left(3+\frac{3}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right) ?\) (A) \(\left(x-3-\frac{3}{\sqrt{2}}\right)^{2}+\left(y-\frac{3}{\sqrt{2}}\right)^{2}=9\) (B) \(\left(x-\frac{3}{\sqrt{2}}\right)^{2}+\left(y-\frac{3}{\sqrt{2}}\right)^{2}=9\) (C) \(x^{2}+(y-3)^{2}=9\) (D) \((x-3)^{2}+y^{2}=9\)

If a circle passes through the points where the lines \(3 k x-2 y-1=0\) and \(4 x-3 y+2=0\) meet the coordinate axes then \(k=\)(A) 1 (B) \(-1\) (C) \(\frac{1}{2}\) (D) \(\frac{-1}{2}\)

Let \(C\) be any circle with centre \((0, \sqrt{2})\). Then, on the circle \(C\), there can be (A) at the most one rational point (B) at the most two rational points (C) at the most three rational points (D) none of these

If the tangents \(P Q\) and \(P R\) are drawn to the circle \(x^{2}+\) \(y^{2}=a^{2}\) from the point \(P\left(x_{1}, y_{1}\right)\), then the equation of the circumcircle of \(\triangle P Q R\) is (A) \(x^{2}+y^{2}-x x_{1}-y y_{1}=0\) (B) \(x^{2}+y^{2}+x x_{1}+y y_{1}=0\) (C) \(x^{2}+y^{2}-2 x x_{1}-2 y y_{1}=0\) (D) none of these

The coordinates of a point \(P\) on the circle \(x^{2}+y^{2}-4 x\) \(-6 y+9=0\) such that \(\angle P O X\) is minimum, where \(O\) is the origin and \(O X\) is the \(x\)-axis, are (A) \(\left(\frac{36}{13}, \frac{15}{13}\right)\) (B) \(\left(\frac{-36}{13}, \frac{15}{13}\right)\) (C) \(\left(\frac{14}{27}, \frac{12}{27}\right)\) (D) none of these

The locus of the centre of a circle which passes through the point \((0,0)\) and cuts off a length \(2 b\) from the line \(x\) \(=c\), is (A) \(y^{2}+2 c x=b^{2}+c^{2}\) (B) \(x^{2}+c x=b^{2}+c^{2}\) (C) \(y^{2}+2 c y=b^{2}+c^{2}\) (D) none of these

If \(P, Q\) is a pair of conjugate points with respect to a circle \(S\), then the circle on \(P Q\) as diameter (A) touches the circle \(S\) (B) cuts the circle \(S\) at an angle \(\frac{\pi}{4}\) (C) cuts the circle \(S\) orthogonally (D) none of these

The common chord of the circle \(x^{2}+y^{2}+8 x+4 y-5=\) 0 and a circle passing through the origin and touching the line \(y=x\), passes through the fixed point (A) \(\left(\frac{5}{12}, \frac{5}{12}\right)\) (B) \(\left(\frac{5}{12}, \frac{-5}{12}\right)\) (C) \(\left(\frac{-5}{12}, \frac{5}{12}\right)\) (D) none of these

If the circles \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}-4 x-6 y+12=0\) cut off equal intercepts on a line which passes through the point \((1,1)\), then the slope of the line is (A) 1 (B) \(-1\) (C) \(\frac{3}{2}\) (D) \(-\frac{3}{2}\)

Consider a curve \(a x^{2}+2 h x y+b y^{2}=1\) and a point \(P\) not on the curve. A line drawn from the point \(P\) intersects the curve at points \(Q\) and \(R\). If the product \(P Q \cdot P R\) is independent of the slope of the line, then the curve is (A) an ellipse (B) a hyperbola (C) a circle (D) none of these

Let \(L_{1}\) be a straight line passing through the origin and \(L_{2}\) be the straight line \(x+y=1\). If the interecpts made by the circle \(x^{2}+y^{2}-x+3 y=0\) on \(L_{1}\) and \(L_{2}\) are equal, then which of the following equations can represent \(L_{1} ?\) (A) \(x+y=0\) (B) \(x-y=0\) (C) \(7 y+2 x=0\) (D) \(x-7 y=0\)

A triangle has two of its sides along the axes. If the third side touches the circle \(x^{2}+y^{2}-2 a x-2 a y+a^{2}=\) 0 , then the equation of the locus of the circumcentre of the triangle is (A) \(2 a(x+y)=2 x y+a^{2}\) (B) \(2 a(x-y)=2 x y+a^{2}\) (C) \(2 a(x+y)=2 x y-a^{2}\) (D) none of these

A point moves such that the sum of the squares of its distances from the sides of a square of side unity is equal to 9 . The locus of the point is a circle such that (A) centre of the circle coincides with that of square (B) centre of the circle is \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (C) radius of the circle is 2 (D) all the above are true

The range of values of \(a\) for which the line \(y+x=0\) bisects twochords drawn froma point \(\left(\frac{1+\sqrt{2} a}{2}, \frac{1-\sqrt{2} a}{2}\right)\) to the circle \(2 x^{2}+2 y^{2}-(1+\sqrt{2} a) x-(1-\sqrt{2} a) y=0\) is \((\) A) \((-\infty,-2) \cup(2, \infty)\) (B) \((-2,2)\) (C) \((2, \infty)\) (D) none of these

The locus of the centres of the circles which touch the two circles \(x^{2}+y^{2}=a^{2}\) and \(x^{2}+y^{2}=4 a x\) externally is (A) \(12 x^{2}-4 y^{2}-24 a x+9 a^{2}=0\) (B) \(12 x^{2}+4 y^{2}-24 a x+9 a^{2}=0\) (C) \(12 x^{2}-4 y^{2}+24 a x+9 a^{2}=0\) (D) none of these

If a circle passes through the points of intersection of the coordinate axes with the lines \(\lambda x-y+1=0\) and \(x-2 y+3=0\), then the value of \(\lambda\) is (A) 2 (B) 1 (C) \(-1\) (D) \(-2\)

A circle touches the line \(y=x\) at \(a\) point \(P\) such that \(O P=4 \sqrt{2}\), where \(O\) is the origin. The circle contains the point \((-10,2)\) in its interior and the length of its chord on the line \(x+y=0\) is \(6 \sqrt{2}\). The equation of the circle is (A) \(x^{2}+y^{2}+18 x-2 y+32=0\) (B) \(x^{2}+y^{2}-18 x-2 y+32=0\) (C) \(x^{2}+y^{2}+18 x+2 y+32=0\) (D) none of these