1997
JEE Advanced
Numerical
IIT-JEE 1997
Let C be any circle with centre $\,\left( {0\, , \sqrt {2} } \right)$. Prove that at the most two rational points can to there on C. (A rational point is a point both of whose coordinates are rational numbers.)
Correct Answer: solve it
1997
JEE Advanced
Numerical
IIT-JEE 1997
The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to circle ${x^2} + {y^2} = 1$ pass through the point........................
Correct Answer: $$\left( {{1 \over 2},{1 \over 4}} \right)$$
1997
JEE Advanced
Numerical
IIT-JEE 1997
For each natural number k, let ${C_k}$ denote the circle with radius k centimetres and centre at the origin. On the circle ${C_k}$, a-particle moves k centimetres in the counter-clockwise direction. After completing its motion on ${C_k}$, the particle moves to ${C_{k + 1}}$ in the radial direction. The motion of the patticle continues in the manner. The particle starts at (1, 0). If the particle crosses the positive direction of the x-axis for the first time on the circle ${C_n}$ then n = ..............
Correct Answer: 7
1996
JEE Advanced
MCQ
IIT-JEE 1996
The angle between a pair of tangents drawn from a point P to the circle ${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0$ is $2\,\alpha $.
The equation of the locus of the point P is
The equation of the locus of the point P is
A.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,4\, = \,0$
B.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, - \,9\,\, = \,0$
C.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, - \,4\,\, = \,0$
D.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, + \,9\,\, = \,0$
1996
JEE Advanced
Numerical
IIT-JEE 1996
Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point $\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - \,\sqrt 2 a} \over 2}} \right)$ to the circle $\,\,2{x^2}\, + \,2{y^2} - (\,1\, + \sqrt 2 a)\,x - (1 - \sqrt 2 a)\,y = 0$.
Correct Answer: $$a\, \in \,]\, - \infty ,\,\, - \,2\,[U]\,\,2,\,\,\infty \,[$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angles DAB and CAB are $\,\alpha \,\,and\,\,\beta $ respectively and the distance between the point A and the mid point of the line segment DC is d, prove that the area of the circle is $${{\pi \,{d^2}\,\,{{\cos }^2}\,\,\alpha } \over {{{\cos }^2}\,\alpha \, + \,{{\cos }^2}\,\beta \, + \,\,2\,\cos \,\,\alpha \,\,\cos \,\beta \,\cos \,\,(\beta - \alpha )\,}}$$
Correct Answer: solve it
1996
JEE Advanced
Numerical
IIT-JEE 1996
The intercept on the line y = x by the circle ${x^2} + {y^2} - 2x = 0$ is AB. Equation of the circle with AB as a diameter is................................
Correct Answer: $${x^2} + {y^2} - x - y = 0$$
1994
JEE Advanced
MCQ
IIT-JEE 1994
The circles ${x^2} + {y^2} - 10x + 16 = 0$ and ${x^2} + {y^2} = {r^2}$ intersect each other in two distinct points if
A.
r < 2
B.
r > 8
C.
2 < r < 8
D.
$2 \le r \le 8$
1993
JEE Advanced
MCQ
IIT-JEE 1993
The locus of the centre of a circle, which touches externally the circle ${x^2} + {y^2} - 6x - 6y + 14 = 0$ and also touches the y-axis, is given by the equation:
A.
${x^2} - 6x - 10y + 14 = 0$
B.
${x^2} - 10x - 6y + 14 = 0$
C.
${y^2} - 6x - 10y + 14 = 0$
D.
${y^2} - 10x - 6y + 14 = 0$
1993
JEE Advanced
Numerical
IIT-JEE 1993
Find the coordinates of the point at which the circles ${x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36$ touch each other. Also find equations common tangests touching the circles in the distinct points.
Correct Answer: $$\left( {{{14} \over 5},{8 \over 5}} \right),\,\,y = 0$$ and $$7y - \,24x\, + \,16\, = 0$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the circle ${x^2}\, + \,{y^2} - \,4x - \,6y - 3 = 0$ cuts the members of the family are concurrent at a point. Find the coordinate of this point.
Correct Answer: $$\left( {2\,,{{23} \over 3}} \right)$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
The equation of the locus of the mid-points of the circle $4{x^2} + 4{y^2} - 12x + 4y + 1 = 0$ that subtend an angle of $2\pi /3$ at its centre is.................................
Correct Answer: $$16{x^2} + 16{y^2} - 48x + 16y + 31 = 0$$
1992
JEE Advanced
MCQ
IIT-JEE 1992
The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle ${x^2} + {y^2} = 9$is
A.
$\left( {{3 \over 2},{1 \over 2}} \right)\,$
B.
$\left( {{1 \over 2},{3 \over 2}} \right)\,$
C.
$\left( {{1 \over 2},{1 \over 2}} \right)\,$
D.
$\left( {{1 \over 2}, - {2^{{1 \over 2}}}} \right)\,$
1992
JEE Advanced
Numerical
IIT-JEE 1992
Let a circle be given by 2x (x - a) + y (2y - b) = 0, $(a\, \ne \,0,\,\,b\, \ne 0)$. Find the condition on a abd b if two chords, each bisected by the x-axis, can be drawn to the circle from $\left( {a,\,\,{b \over 2}} \right)$.
Correct Answer: $${a^2}\, > \,2\,{b^2}$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the circles.
Correct Answer: <br> $${x^2}\, + \,{y^2} + 6x\, + 2y - 15\, = 0$$ and
<br> $${x^2}\, + \,{y^2} - 10x\, - 10y + 25\, = 0$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
If a circle passes through the points of intersection of the coordinate axes with the lines $\lambda \,x - y + 1 = 0$ and x - 2y + 3 = 0, then the value of $\lambda $ = .........
Correct Answer: 2
1990
JEE Advanced
Numerical
IIT-JEE 1990
A circle touches the line y = x at a point P such that OP = ${4\sqrt 2 \,}$, 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 ${6\sqrt 2 \,}$. Determine the equation of the circle.
Correct Answer: $${x^2}\, + {y^2} + \,18x\, - 2y\, + 32\, = 0$$
1989
JEE Advanced
MCQ
IIT-JEE 1989
The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle of area 154 sq. units. Then the equation of this circle is
A.
${x^2} + {y^2} + 2x - 2y = 62$
B.
${x^2} + {y^2} + 2x - 2y = 47$
C.
${x^2} + {y^2} - 2x + 2y = 47$
D.
${x^2} + {y^2} - 2x + 2y = 62$c
1989
JEE Advanced
MCQ
IIT-JEE 1989
If the two circles ${(x - 1)^2} + {(y - 3)^2} = {r^2}$ and ${x^2} + {y^2} - 8x + 2y + 8 = 0$ intersect in two distinct points, then
A.
2 < r < 8
B.
r < 2
C.
r = 2
D.
r > 2
1989
JEE Advanced
Numerical
IIT-JEE 1989
If $\left( {{m_i},{1 \over {{m_i}}}} \right),\,{m_i}\, > \,0,\,i\, = 1,\,2,\,3,\,4$ are four distinct points on a circle, then show that ${m_1}\,{m_2}\,{m_3}\,{m_4}\, = 1$
Correct Answer: solve it
1989
JEE Advanced
Numerical
IIT-JEE 1989
The area of the triangle formed by the positive x-axis and the normal and the tangent to the circle ${x^2} + {y^2} = 4\,\,at\,\,\left( {1,\sqrt 3 } \right)$ is,..................
Correct Answer: $${2\sqrt 3 }$$ sq unit
1989
JEE Advanced
MCQ
IIT-JEE 1989
The line x + 3y = 0 is a diameter of the circle ${x^2} + {y^2} - 6x + 2y = 0\,$.
A.
TRUE
B.
FALSE
1988
JEE Advanced
MCQ
IIT-JEE 1988
If a circle passes through the point (a, b) and cuts the circle ${x^2}\, + \,{y^2}\, = \,{k^2}$ orthogonally, then the equation of the locus of its centre is
A.
$2\,ax\, + \,2\,by\, - \,({a^2}\, + \,{b^2}\, + \,\,{k^2})\, = \,0$
B.
$2\,ax\, + \,2\,by\, - \,({a^2}\, - \,\,{b^2}\, + \,\,{k^2})\, = \,0$
C.
${x^2}\, + \,{y^2}\, - \,3\,\,ax\, + \,4\,by\, + \,\,({a^2}\, + \,\,{b^2}\, - \,\,{k^2})\, = \,0$
D.
${x^2}\, + \,{y^2}\, - \,2\,\,ax\, - \,4\,by\, + \,\,({a^2}\, - \,\,{b^2}\, - \,\,{k^2})\, = \,0$.
1988
JEE Advanced
MSQ
IIT-JEE 1988
The equations of the tangents drawn from the origin to the circle ${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$, are
A.
x = 0
B.
y = 0
C.
$({h^2}\, - \,{r^2})\,x - \,\,2rhy\, = \,0$
D.
$({h^2}\, - \,{r^2})\,x + \,\,2rhy\, = \,0$
1988
JEE Advanced
Numerical
IIT-JEE 1988
If the circle ${C_1}:{x^2} + {y^2} = 16$ intersects another circle ${C_2}$ of radius 5 in such a manner that common chord is of maximum lenght and has a slope equal to 3/4, then the coordinates of the centre of ${C_2}$ are.............................
Correct Answer: $$\,\left( { - {9 \over 5},{{12} \over 5}} \right)\,\,or\,\left( {{9 \over 5},\, - {{12} \over 5}} \right)$$
1987
JEE Advanced
Numerical
IIT-JEE 1987
Let a given line $L_1$ intersects the x and y axes at P and Q, respectively. Let another line $L_2$, perpendicular to $L_1$, cut the x and y axes at R and S, respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.
Correct Answer: solve it
1987
JEE Advanced
Numerical
IIT-JEE 1987
The circle ${x^2}\, + \,{y^2} - \,4x\, - 4y + \,4 = 0$ is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is $x\, + \,y\, - xy\, + k\,{\left( {{x^2}\, + \,{y^2}} \right)^{1/2}} = 0$. Find k.
Correct Answer: k = 1
1987
JEE Advanced
Numerical
IIT-JEE 1987
The area of the triangle formed by the tangents from the point (4, 3) to the circle ${x^2} + {y^2} = 9$ and the line joining their points of contact is...................
Correct Answer: $${{192} \over {25}}$$
1986
JEE Advanced
Numerical
IIT-JEE 1986
Lines 5x + 12y - 10 = 0 and 5x - 12y - 40 = 0 touch a circle $C_1$ of diameter 6. If the centre of $C_1$ lies in the first quadrant, find the equation of the circle $C_2$ which is concentric with $C_1$ and cuts intercepts of length 8 on these lines.
Correct Answer: $${x^2}\, + \,{y^2} - \,10x\, - 4y + \,4 = 0\,$$
1986
JEE Advanced
Numerical
IIT-JEE 1986
The equation of the line passing through the points of intersection of the circles $3{x^2} + 3{y^2} - 2x + 12y - 9 = 0$ and ${x^2} + {y^2} - 6x + 2y - 15 = 0$ is..............................
Correct Answer: 10x - 3y - 18 = 0
1986
JEE Advanced
Numerical
IIT-JEE 1986
From the point A(0, 3) on the circle ${x^2} + 4x + {(y - 3)^2} = 0$, a chord AB is drawn and extended to a point M such that AM = 2AB. The equation of the locus of M is..........................
Correct Answer: $${x^2} + {y^2} + 8x - 6y + 9 = 0$$
1985
JEE Advanced
Numerical
IIT-JEE 1985
From the origin chords are drawn to the circle ${(x - 1)^2} + {y^2} = 1$. The equation of the locus of the mid-points of these chords is.............
Correct Answer: $${x^2} + {y^2} - x = 0$$
1985
JEE Advanced
Numerical
IIT-JEE 1985
Let ${x^2} + {y^2} - 4x - 2y - 11 = 0$ be a circle. A pair of tangentas from the point (4, 5) with a pair of radi from a quadrilateral of area............................
Correct Answer: 8 sq unit
1985
JEE Advanced
MCQ
IIT-JEE 1985
No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices $\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt 3 } \right),\,\,\left( {3,\sqrt 3 } \right)$.
A.
TRUE
B.
FALSE
1984
JEE Advanced
MCQ
IIT-JEE 1984
The locus of the mid-point of a chord of the circle ${x^2} + {y^2} = 4$ which subtends a right angle at the origin is
A.
x + y = 2
B.
${x^2} + {y^2} = 1$
C.
${x^2} + {y^2} = 2$
D.
$x + y $=1
1984
JEE Advanced
Numerical
IIT-JEE 1984
The abscissa of the two points A and B are the roots of the equation ${x^2}\, + \,2ax\, - {b^2} = 0$ and their ordinates are the roots of the equation ${x^2}\, + \,2px\, - {q^2} = 0$. Find the equation and the radius of the circle with AB as diameter.
Correct Answer: $${x^2}\, + \,{y^2} + \,2ax\, + 2py\, - {b^2}\, - {q^2} = 0,\,\,\,\sqrt {{a^2}\, + \,{p^2} + {b^2}\, + \,{q^2}} $$
1984
JEE Advanced
Numerical
IIT-JEE 1984
The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle. The radius of this circle is ........................................
Correct Answer: $$\,{\raise0.5ex\hbox{$\scriptstyle 3$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 4$}}$$
1983
JEE Advanced
MCQ
IIT-JEE 1983
The equation of the circle passing through (1, 1) and the points of intersection of ${x^2} + {y^2} + 13x - 3y = 0$ and $2{x^2} + 2{y^2} + 4x - 7y - 25 = 0$ is
A.
$4{x^2} + 4{y^2} - 30x - 10y - 25 = 0$
B.
$4{x^2} + 4{y^2} + 30x - 13y - 25 = 0$
C.
$4{x^2} + 4{y^2} - 17x - 10y + 25 = 0$
D.
none of these
1983
JEE Advanced
MCQ
IIT-JEE 1983
The centre of the circle passing through the point (0, 1) and touching the curve $\,y = {x^2}$ at (2, 4) is
A.
$\left( {{{ - 16} \over 5},{{ - 27} \over {10}}} \right)$
B.
$\left( {{{ - 16} \over 7},{{53} \over {10}}} \right)$
C.
$\left( {{{ - 16} \over 5},{{53} \over {10}}} \right)$
D.
none of these
1983
JEE Advanced
Numerical
IIT-JEE 1983
Through a fixed point (h, k) secants are drawn to the circle $\,{x^2}\, + \,{y^2} = \,{r^2}$. Show that the locus of the mid-points of the secants intercepted by the circle is $\,{x^2}\, + \,{y^2} $ = $hx + ky$.
Correct Answer: solve it
1983
JEE Advanced
Numerical
IIT-JEE 1983
The point of intersection of the line 4x - 3y - 10 = 0 and the circle ${x^2} + {y^2} - 2x + 4y - 20 = 0$ are ........................and ...................
Correct Answer: (4, 2), (- 2, - 6)
1982
JEE Advanced
Numerical
IIT-JEE 1982
If A and B are points in the plane such that PA/PB = k (constant) for all P on a given circle, then the value of k cannot be equal to ..........................................
Correct Answer: 1
1981
JEE Advanced
Numerical
IIT-JEE 1981
Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2.
Correct Answer: $${x^2}\, + \,{y^2} + \,\,2\,\left( {10\,\, \pm \,\,\sqrt {54} } \right)\,x\, + \,\,55\,\, \pm \,\,\sqrt {54} = \,0$$
1981
JEE Advanced
Numerical
IIT-JEE 1981
Let A be the centre of the circle ${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$. Suppose that the tangents at the points B (1, 7) and D (4. - 2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.
Correct Answer: 72 sq units
1980
JEE Advanced
MCQ
IIT-JEE 1980
Two circles ${x^2} + {y^2} = 6$ and ${x^2} + {y^2} - 6x + 8 = 0$ are given. Then the equation of the circle through their points of intersection and the point (1, 1) is
A.
${x^2} + {y^2} - 6x + 4 = 0$
B.
${x^2} + {y^2} - 3x + 1 = 0$
C.
${x^2} + {y^2} - 4y + 2 = 0$
D.
none of these
1980
JEE Advanced
MCQ
IIT-JEE 1980
A square is inscribed in the circle ${x^2} + {y^2} - 2x + 4y + 3 = 0$. Its sides are parallel to the coordinate axes. The one vertex of the square is
A.
$\left( {1 + \sqrt {2,} - 2} \right)$
B.
$\,\left( {1 - \sqrt {2}, - 2} \right)$
C.
$\,\,\left( {1, - 2 + \sqrt 2 } \right)$
D.
none of these
1978
JEE Advanced
Numerical
IIT-JEE 1978
Find the equation of the circle whose radius is 5 and which touches the circle ${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$ at the point (5, 5).
Correct Answer: $${x^2}\, + \,{y^2}\, - \,18x\,\, - 16y\, + 120 = 0\,$$