Home
Class 12
MATHS
The centres of those circles which touch...

The centres of those circles which touch the circle, `x^(2)+y^(2)-8x-8y-4=0`, externally and also touch the x-axis, lie on

A

a circle

B

an ellipse which is not a circle

C

a hyperbola0

D

a parabola

Text Solution

Verified by Experts

Given equation ofcircle is
`x^(2)+y^(2)-8x-8y-4=0`, whose centre is C(4, 4) and radius `=sqrt(4^(2)+4^(2)+4) = sqrt36 = 6`
Let the centre of required circle be `C_(1)(x, y)`. Now, as it touch the X-axis, therefore its radius ` = |y|`. Also, it touch the circle
`x^(2)+y^(2)-8x-8y-4=0 " therefore " "CC"_(1)=6+|y|`
`rArr sqrt((x-4)^(2)+(y-4)^(2))=6+|y|`
`rArr x^(2)+16-8x+y^(2)+16-8y=36+y^(2)+12|y|`
`rArr x^(2)-8x-8y+32=36+12|y|`
`rArr x^(2)-8x-8y-4=12|y|`
Case I if `ygt0`, then we have
`x^(2)-8x-8y-4=12y`
`rArr x^(2)-8x-20y-4=0`
`rArr (x-4)^(2)-20=20y`
`rArr (x-4)^(2)=20(y+1)`, which is a parabola
Case II if `ylt 0`, then we have
`x^(2)-8x-8y-4=-12y`
`rArr x^(2)-8x-8y-4+12y=0`
`rArr x^(2)-8x+4y-4=0`
`rArr x^(2)-8x-4=-4y`
`rArr (x-4)^(2)=20-4y`
`rArr(x-4)^(2)=-4(y-5)`, which is again a parabola .
Promotional Banner

Similar Questions

Explore conceptually related problems

The locus of the centres of the circles, which touch the circle, x^(2)+y^(2)=1 externally, also touch the Y-axis and lie in the first quadrant, is

The locus of the centre of a circle which touches externally the circle x^2 + y^2-6x-6y+14 = 0 and also touches Y-axis, is given by the equation (a) x^2-6x-10y+14 = 0 (b) x^2-10x-6y + 14 = 0 (c) y^2+6x-10y+14-0 (d) y^2-10x-6y + 14 = 0

The locus of the centres of circles which touches (y-1)^(2)+x^(2)=1 externally and also touches X-axis is

Consider the locus of center of the circle which touches the circle x^(2)+y^(2)=4 externally and the line x=4. The distance of the vertex of the locus from the otigin is __________ .

Find the centre and the radius of the circle x^(2)+y^(2)+8x+10y-8=0 .

Find the equation of the circle whose radius is 5a n d which touches the circle x^2+y^2-2x-4y-20=0 externally at the point (5,5)dot

The minimum area of circle which touches the parabolas y=x^(2)+1andy^(2)=x-1 is

Find the locus of the center of the circle touching the circle x^2+y^2-4y=4 internally and tangents on which from (1, 2) are making of 60^0 with each other.

Find the centre and radius of circles. x^(2)+y^(2)-4x-8y-45=0 .

The centre of the smallest circle touching the circles x^2+ y^2-2y -3=0 and x^2 +y^ 2-8x -18y +93= 0 is: