﻿ Let C­1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1 touches C1 internally and C2 externally. Identify the locus of the centre of C. : Kaysons Education

Let C­1 and C2 be Two Circles With C2 lying Inside C1. A Circle C lying Inside C1 touches C1 internally And C2 externally. Identify The Locus Of The Centre Of C.

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Question

Solution

Correct option is

(ab) and (0, 0)

Let the given circles C1 and C2 have centres O1 and O2 with radii r1 andr2 respectively. Let centre of circle C is at O radius is r.

Which is greater than O1O2 as O1O2 < r1 + r2.

Alternative Method:

Adding (1) and (2) we get

Which represents an ellipse with foci are at (ab) and (0, 0).

SIMILAR QUESTIONS

Q1

P is a variable on the line y = 4. Tangents are drawn to the circle x2 + y2= 4 from P to touch it at A and B. The perpendicular PAQB is completed. Find the equation of the locus of Q.

Q2

Find the condition on abc such that two chords of the circle

x2 + y2 – 2ax – 2by + a2 + b2 – c2 = 0

passing through the point (ab + c) are bisected by the line y = x.

Q3

Find the limiting points of the circles

(x2 + y2 + 2gx + c) + λ(x2 + y2 + 2fy + d) = 0

Q4

The circle x2 + y2 – 4x – 8y + 16 = 0 rolls up the tangent to it at  by 2 units, assuming the x-axis as horizontal, find the equation of the circle in the new position.

Q5

Find the equation of the circle of minimum radius which contains the three circles

x2 – y2 – 4y – 5 = 0

x2 + y2 + 12x + 4y + 31 = 0

and         x2 + y2 + 6x + 12y + 36 = 0

Q6

Find the equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius.

Q7

If the line x cos α + y sin α = p cuts the circle x2 + y2 = a2 in M and N, then show that the circle, whose diameter is MN, is

Q8

A line meets the co-ordinate axes at A and B. A circle is circumscribed about the triangle OAB. If the distance of the points A and B from the tangent at O, the origin, to the circle are m and n respectively, find the equation of the circle.

Q9

Tangents PQPR are drawn to the circle x2 + y2 = 36 from the point P(–8, 2) touching the circle at QR respectively. Find the equation of the circumcircle of the âˆ†PQR.

Q10

The circle x2 + y2 – 4x – 4y + 4 = 0 is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is x + y – xy + k(x2 + y2)1/2 = 0. Find k.