# Family of Circles

## Trending Questions

**Q.**The circle passing through the point (−1, 0) and touching the y-axis at (0, 2) also passes through the point:

- (−52, 2)
- (3, 5)
- (−4, 0)
- (−32, 0)

**Q.**Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C2 meet the side AB at E. If the length of EB is α+√3β where α, β are integers, then α+β is equal to

**Q.**The equation arg (z−1z+1)=π4 represents a circle with:

- centre at (0, 0) and radius √2
- centre at (0, 1) and radius √2
- centre at (0, −1) and radius √2
- centre at (0, 1) and radius 2

**Q.**Equation of the circle passing through the point (1, 1) and point of intersection of circles x2+y2=6 and x2+y2−6x+8=0 is

- x2+y2−6x+4=0
- x2+y2−3x+1=0
- x2+y2−4y+2=0
- x2+y2−3y+1=0

**Q.**The radius of the smallest circle which touches the straight-line 3x−y=6 at (1, −3) and also touches the line y=x is

(Compute upto one place of decimal only)

- 2 units
- 1.5 units
- 1.8 units
- 1.2 units

**Q.**The equation(s) of the circle passing through the points of intersection of the circles x2+y2−2x−4y−4=0, x2+y2−10x−12y+40=0 and having radius 4 units is/are

- 2x2+2y2−18x−22y+69=0
- x2+y2+28x−22y+19=0
- x2+y2−2y−15=0
- x2+y2+2x−15=0

**Q.**The radius of the circle touching the coordinate axes and the line 3x+4y=12 may be

- 6
- 2
- 3
- 1

**Q.**Find the vector →a of magnitude 5√2, making an angle of π4 with x-axis, π2 with y-axis and an angle θ with z-axis.

**Q.**A variable straight line AB divides the circumference of the circle x2+y2=25 in the ratio 1:2. If a tangent CD is drawn to the smaller arc parallel to AB, such that ABCD is a rectangle (as shown in the figure), then locus of C and D is

- x2+y2=1754
- x2+y2=36
- x2+y2=40
- x2+y2=20

**Q.**Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval

- 0<k<12
- k≥12
- −12≤k≤12
- k≤12

**Q.**Find the equation of a family of circles touching the lines x2−y2+2y−1=0. (where h and k are parameters)

- x2+y2−2ky+k22+k−12=0
- x2+y2−2hx−2y+h22+1=0
- x2+y2−2hx+2y+h22=0
- x2+y2−2ky+k22=0

**Q.**S is circle having center at (0, a) and radius b(b<a). A is a variable circle centered at (α, 0) and touching circle S, meets the x− axis at M and N. If a point P on the y− axis such that ∠MPN is independent from α, then

- ∠MPN=cos−1(ba)
- P≡(0, √a2−b2)
- ∠MPN=−cos−1(ba)
- P≡(0, −√a2−b2)

**Q.**Consider the circle x2+y2−10x−6y+30=0. Let O be the centre of the circle and tangents at A(7, 3) and B(5, 1) meet at C. Let S=0 represents the family of circles passing through A and B. Then ,

- The area of quadrilateral OACB is 4
- The radical axis for the family of circles S=0 is x+y=10
- The smallest possible circle of the family S=0 is x2+y2−12x−4y+38=0
- The coordinates of point C are (7, 1)

**Q.**Consider the family of circles x2+y2−2x−2λy−8=0 passing through two fixed points A and B. Then the distance between the points A and B is

units

**Q.**If the curves ax2+4xy+2y2+x+y+5=0 and ax2+6xy+5y2+2x+3y+8=0 intersect at four concyclic points, then the value of a is

- 4
- −4
- −6
- 6

**Q.**Five circles C1, C2, C3, C4, C5 with radii r1, r2, r3, r4, r5 respectively (r1<r2<r3<r4<r5) be such that Ci and Ci+1 touch each other externally for all i=1, 2, 3, 4. If all the five circles touch two straight lines L1 and L2 and r1=2 and r5=32, then r3 is

**Q.**If the curves ax2+4xy+2y2+x+y+5=0 and ax2+6xy+5y2+2x+3y+8=0 intersect at four concyclic points then the value of a is

- 4
- -6
- -4
- 6

**Q.**Equation of the circle passing through the point (1, 1) and point of intersection of circles x2+y2=6 and x2+y2−6x+8=0 is

- x2+y2−6x+4=0
- x2+y2−3x+1=0
- x2+y2−4y+2=0
- x2+y2−3y+1=0

**Q.**Find the angles between the lines

x−√3y=1

**Q.**If a circle passes through the point (1, 2) and cuts the circle x2+y2=4 orthogonally then equation of the locus of its centre is the straight line 2x+4y+9=0.

**Q.**At the point (2, 3) on the curve y=x3−2x+1, the gradient of the curve increases k times as fast as its abscissa, Then the value of k is

**Q.**Conisder the circle x2+y2−10x−6y+30=0. Let O be the centre of the circle and tangent at A(7, 3) and B(5, 1) meet at C. Let S=0 represents family of circles passing through A and B, then

- Area of quadrilateral OACB=4 sq. units
- The radical axis for the family of circles S=0 is x+y=10
- The smallest possible circle of the family S=0 is x2+y2−12x−4y+38=0
- The coordinates of point C are (7, 1)

**Q.**The line x=y touches a circle at the point (1, 1). If the circle also passes through the point (1, −3) then its radius is:

- 3
- 2√2
- 2
- 3√2

**Q.**Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval

- 0<k<12
- k≥12
- −12≤k≤12
- k≤12

**Q.**The area common to the circles r=a√2 and r=2acosθ is:

- a2π2
- a2π
- a2(π+1)
- a2(π−1)

**Q.**The range of value of P such that the angle θ between the pair of tangents drawn from the point (p.0) to the circle x2+y2=1 lies in (π3, π), is

- (−3, −2)∪(2, 3)
- (0, 2)
- (−2, −1)∪(1, 2)
- None of these

**Q.**Consider the family of circles x2+y2−2x−2λy−8=0 passing through two fixed points A and B. Then the distance between the points A and B is

units

**Q.**

Choose the correct answer in the following question:

The point on the curve 9y2=x3, where the normal to the curve makes equal intercepts with the axes is

(a) (4, ±83) (b) (4, −83)

(c) (4, ±38) (d) (±4, 38)

**Q.**The line x=y touches a circle at the point (1, 1). If the circle also passes through the point (1, −3), then its radius(in units) is

- 2
- 3√2
- 3
- 2√2

**Q.**Find the equation of the circle passing through the points of intersection of the circles x2+y2−2x−4y−4=0 and x2+y2−10x−12y+40=0 and whose radius is 4