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Mere Teacher · Previous Year Questions
JEE Main Mathematics · Circle · Set 1 (Q1-30)
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1.
If the circles x2+y2+5Kx+2y+K=0 and 2(x2+y2)+2Kx+3y−1=0,(KϵR), intersect at the points P and Q, then the line 4x+5y−K=0 passes through P and Q, for :
(1)infinitely many values of K
(2)no value of K
(3)exactly two values of K
(4)exactly one value of K
2.
The two circles x2+y2=ax and x2+y2=c2(c>0) touch each other if:
(1)2|a|=c
(2)|a|=c
(3)a=2c
(4)|a|=2c
3.
If the points of intersection of the ellipses x2+2y2−6x−12y+23=0 and 4x2+2y2−20x−12y+35=0 lie on a circle of radius r and centre (a,b), then the value of ab+18r2 is
(1)52
(2)53
(3)51
(4)55
4.
Let C:x2+y2=4 and C’:x2+y2−4λx+9=0 be two circles. If the set of all values of λ so that the circles C and C’ intersect at two distinct points, is R−[a,b], then the point (8a+12,16b−20) lies on the curve :
(1)x2+2y2−5x+6y=3
(2)5x2−y=−11
(3)x2−4y2=7
(4)6x2+y2=42
5.
Let the circle x2+y2=4 intersect x-axis at the points A(a,0),a>0 and B(b,0). Let P(2cosα,2sinα), 0<α<2π and Q(2cosβ,2sinβ) be two points such that (α−β)=2π. Then the point of intersection of AQ and BP lies on :
(1)x2+y2−4y−4=0
(2)x2+y2−4x−4y=0
(3)x2+y2−4x−4=0
(4)x2+y2−4x−4y−4=0
6.
Let the lengths of intercepts on X -axis and y -axis made by the circle x2+y2+ax+2ay+c=0(a<0) be 22 and 25, respectively. Thenthe shortest distance from origin to a tangent to this circle which is perpendicular to the line x+2y=0, iseuqalto :
(1)11
(2)7
(3)6
(4)10
7.
If a tangent to the circle x2+y2=1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:
(1)x2+y2−4x2y2=0
(2)x2+y2−2xy=0
(3)x2+y2−16x2y2=0
(4)x2+y2−2x2y2=0
8.
If the line x−2y=m(m∈Z) intersects the circle x2+y2=2x+4y at two distinct points, then the number of possible values of m are
(1)8
(2)9
(3)10
(4)11
9.
If the circles x2+y2−16x−20y+164=r2 and (x−4)2+(y−7)2=36 intersect at two distinct points, then:
(1)r> 11
(2)0 < r < 1
(3)r = 11
(4)1 < r < 11
10.
The equation of the circle passing through the point (1,0) and (0,1) and having the smallest radius is -
(1)x2+y2−2x−2y+1=0
(2)x2+y2−x−y=0
(3)x2+y2+2x+2y−7=0
(4)x2+y2+x+y−2=0
11.
The equation(s) of the common tangent(s) to the circle x2+y2−6x=0 and the parabola, y2=4xis:
(1)23y=12x+1
(2)3y=x+3
(3)23y=−x−12
(4)3y=3x+1
12.
The number of common tangents to the circles x2+y2−4x−6y−12=0 and x2+y2+6x+18y+26=0, is
(1)1
(2)2
(3)3
(4)4
13.
Let PQ and MN be two straight lines touching the circle x2+y2−4x−6y−3=0 at the points A and B respectively. Let O be the centre of the circle and ∠AOB=π/3. Then the locus of the point of intersection of the lines PQ and MN is :
(1)x2+y2−12x−18y−25=0
(2)3(x2+y2)−18x−12y+25=0
(3)3(x2+y2)−12x−18y−25=0
(4)x2+y2−18x−12y−25=0
14.
The equation of the circle whose centre lies on the line x−4y=1 and which passes through the points (3,7) and (5,5) is
(1)x2+y2+6x+2y−90=0
(2)x2+y2+6x−2y+90=0
(3)x2+y2+6x+2y+90=0
(4)x2+y2−6x+2y−90=0
15.
Choose the correct statement about two circles whose equations are given below: x2+y2−10x−10y+41=0 x2+y2−22x−10y+137=0
(1)circles have same centre
(2)circles have no meeting point
(3)circles have only one meeting point
(4)circles have two meeting points
16.
The equation of tangents to the circle x2+y2=4 which are parallel to x+2y+3=0 are
(1)x−2y=±25
(2)x−2y=±2
(3)x+2y=±23
(4)x+2y=±25
17.
The points of intersection of the line ax+by=0(a=b) and the circle x2+y2−2x=0 are A(α,0) and B(1,β). The image of the circle with AB as a diameter in the line x+y+2=0 is
(1)x2+y2+5x+5y+12=0
(2)x2+y2+3x+5y+8=0
(3)x2+y2+3x+3y+4=0
(4)x2+y2−5x−5y+12=0
18.
Let the set of all values of r, for which the circles (x+1)2+(y+4)2=r2 and x2+y2−4x−2y−4=0 intersect at two distinct points be the interval (α,β). Then αβ is equal to
(1)21
(2)24
(3)20
(4)25
19.
Let a circle of radius 4 pass through the origin O , the points A(−3a,0) and B(0,−2b), where a and b are real parameters and ab=0. Then the locus of the centroid of △OAB is a circle of radius
(1)337
(2)38
(3)311
(4)35
20.
Let the circle C1:x2+y2−2(x+y)+1=0 and C2 be a circle having centre at (−1,0) and radius 2 . If the line of the common chord of C1 and C2 intersects the y-axis at the point P , then the square of the distance of P from the centre of C1 is :
(1)2
(2)1
(3)4
(4)6
21.
The circle on the chord xcosα+ysinα=p of the circle x2+y2=a2 as diameter has the equation x2+y2−a2−Np(xcosα+ysinα−p)=0 Find N.
22.
If the point (2,0),(0,1),(4,5) and (0,c) are con-cyclic, then integral value of c is equal to
23.
The two points A and B in a plane are such that for all points P lies on circle satisfied PBPA=k, then k will not be equal to (Ans least positive integer)
24.
Let PQbe a diameter of the circle x2+y2=9. If αand β are the lengths of the perpendiculars from P and Q on the straight line, x+y=2 respectively, then the maximum value of αβis
25.
The tangents are drawn from the point (4,5) to the circle x2+y2−4x−2y−11=0. The area of quadrilateral formed by these tangents and radii, is
26.
If y=2x is a chord of the circle x2+y2−10x=0, then the equation of the circle of which this chord is a diameter, is x2+y2−Ax−By=0. Find A+B.
27.
If the straight line ax+by=2;a,b=0 touches the circle x2+y2−2x=3 and is normal to the circle x2+y2−4y=6then the value of b is
28.
A circle lies in the second quadrant and touches both the axes. If the radius of the circle be 4, then its equation is x2+y2+8x+Ay+B=0. Find A+B.
29.
If the curves, x2−6x+y2+8=0 and x2−8y+y2+16−k=0,(k>0)touch each other at a point, then the largest value of k is
30.
The circles x2+y2−10x+16=0 and x2+y2=r2 intersect each other in two distinct points, if A<r<B. Find A+B
