Question

If two circles which pass through the points (0, a) and (0, –a) cut each other orthogonally and touch the straight line 

y = mx + c, then

Solution

Correct option is

c2 = a2 (2 + m2)

Equation of a family of circles through (0 , a) and (0, –a) is x2 + y2 + 2λax – a2 = 0. If two members are for λ = λ1 and λ = λ2 then since they intersect orthogonally 1λ2a2 = –2a2 ⇒ λ1λ2 = –1

Since the two circles touch the line y = mx + c   

                 

  

SIMILAR QUESTIONS

Q1

An equation of the circle through (1, 1) and the points of intersection ofx2 + y2 + 13x – 3y = 0 and 2x2 + 2y2 + 4x – 7y – 25 = 0 is   

Q2

The abscissae of two points A and B are the roots of the equation x2 + 2ax – b2 = 0, and their ordinates are the roots of the equation x2 + 2px –q2 = 0. The radius of the circle with AB as diameter is  

Q3

The locus of the point of intersection of the tangent to the circle x = r cos θ, y = r sin θ at points whose parametric angles differ by

 is 

Q4

The locus of a point which moves such that the tangents from it to the two circles x+ y2 – 5x – 3 = 0 and 3x2 + 3y2 + 2x + 4y – 6 = 0 are equal is 

Q5

If the two circles x2 + y2 + 2gx + 2fy = 0 and x2 + y2 + 2g1x + 2f1y = 0 touch each other, then

Q6

If two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 cut the coordinates axes in concyclic points, then 

Q7

The locus of the point which moves in a plane so that the sum of the squares of its distances from the lines ax + by + c = 0 and

bx – ay + d = 0 is r2, is a circle of radius.  

 

Q8

The locus of the centre of the circle passing through the origin O and the points of intersection of any line through (ab) and the coordinates axis is a

Q9

Four distinct point (1, 0), (0, 1), (0, 0) and (tt) are concyclic for

Q10

The coordinates of two point P and Q are (2, 3) and (3, 2) respectively. Circles are described on OP and OQ as diameters; O being the origin, then length of the common chord is