Question

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

Solution

Correct option is

From the fig it is clear that A is the centre of the in-circle of the trianglePQR formed by the lines joining the centres of the given circles. We have PQ = 5 + 4 = 9, QR = 4 + 3 = 7, RP = 3 + 5 = 8, distance of from the points of contact is the radius r of the in-circle of triangle PQR. From trigonometry r = Δ/s where Δ is the area and 2s is the perimeter of the triangle PQR

   

                                                                                   

SIMILAR QUESTIONS

Q1

Locus of the mid-points to the chords of the circle x2 + y2 = 4 which subtend a right angle at the centre is

Q2

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

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

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