## 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

2*x*^{2} + 2*y*^{2} – *x* – 7*y* = 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

x^{2} + y^{2} - 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

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

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

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

If a circle passes through the point (*a*, *b*) and cuts the circle *x*^{2} + *y*^{2} = *k*^{2} orthogonally, equation of the locus of its centre is

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

*x*^{2} + *y*^{2} – 4*x* + 2*y* + 4 = 0 orthogonally, is

If *O* is the origin and *OP*, *OQ* are distinct tangents to the circle *x*^{2} + *y*^{2} + 2*gx* + 2*fy* + *c* = 0, the circumcentre of the triangle *OPQ* is

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

If *OA* and *OB* are the tangents from the origin to the circle *x*^{2} + *y*^{2} + 2*gx* + 2*fy* + *c* = 0, and *C* is the centre of the circle, the area of the quadrilateral *OACB* is

The angle between a pair of tangents drawn from a point *P* to the circle *x*^{2} + *y*^{2} + 4*x* – 6*y* + 9 sin^{2}α + 13 cos^{2}α = 0 is 2α. The equation of the locus of the point *P* is

If a line segment *AM* = *a* moves in the plane *XOY* remaining parallel to*OX* so that the left end point *A* slides along the circle *x*^{2} + *y*^{2} = *a*^{2}, the locus of *M* is