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> $${x^2}\, + \,{y^2} - 10x\, - 10y + 25\, = 0$$
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
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
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
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: $${x^2}\, + \,{y^2} - \,10x\, - 4y + \,4 = 0\,$$
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: $${x^2}\, + \,{y^2} + \,2ax\, + 2py\, - {b^2}\, - {q^2} = 0,\,\,\,\sqrt {{a^2}\, + \,{p^2} + {b^2}\, + \,{q^2}} $$
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
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
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: $${x^2}\, + \,{y^2}\, - \,18x\,\, - 16y\, + 120 = 0\,$$
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
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: $${x^2} + {y^2} - x - y = 0$$
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: $$16{x^2} + 16{y^2} - 48x + 16y + 31 = 0$$
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
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
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
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
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
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
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
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
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