Question

Equation of a circle through the origin and belonging to the co-axial system, of which the limiting points are (1, 2), (4, 3) is

Solution

Correct option is

2x2 + 2y2 – x – 7y = 0

Since the limiting points of a system of co-axial circles are the point circles (radius being zero), two members of the system are

and (x-4)2 + (y-3)2 = 0 

x2 + y2 - 8x -6y +25 = 0

 

The co-axial system of circles with these as members is  

  

It passes through the origin if 5 + 25λ = 0  

or                                              λ = –(1/5),

which gives the equation of the required circle as 

  

SIMILAR QUESTIONS

Q1

A circle C touches the x-axis and the circle x2 + (y – 1)2 = 1externally, then locus of the centre of the circle is given by

Q2

Three circles with radii 3 cm, 4 cm and 5 cm touch each other externally. If A is the point of intersection of tangents to these circles at their points of contact, then the distance of A from the points of contact is

Q3

A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the diameter of the circle is

Q4

 

If a circle passes through the point (ab) and cuts the circle x2 + y2 = k2 orthogonally, equation of the locus of its centre is

Q5

Equation of the circle which passes through the origin, has its centre on the line x + y = 4 and cuts the circle

x2 + y2 – 4x + 2y + 4 = 0 orthogonally, is

Q6

If O is the origin and OPOQ are distinct tangents to the circle x2 + y2 + 2gx + 2fy + c = 0, the circumcentre of the triangle OPQ is

Q7

The circle passing through the distinct points (1, t), (t, 1) and (tt) for all values of t, passes through the point

Q8

If OA and OB are the tangents from the origin to the circle x2 + y2 + 2gx + 2fy + c = 0, and C is the centre of the circle, the area of the quadrilateral OACB is 

Q9

The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x – 6y + 9 sin2α + 13 cos2α = 0 is 2α. The equation of the locus of the point P is

Q10

If a line segment AM = a moves in the plane XOY remaining parallel toOX so that the left end point A slides along the circle x2 + y2 = a2, the locus of M is