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1.

Find the equation of the circle which touches thelines `4x-3y+10=0a n d4x-3y-30=0`and whose centre lies on the line `2x+y=0.`

Answer» `4x-3y+10=0`
`4x-3y-30=0`
solving the line
`(2x+y=0)`and`(4x-2y+10=0)`
`x=-1,y=2`
Solving the line
`(4x-3y-3=0)`and`(2x+y=0)`
`x=3,y=-6`
Distance between parallel line=d
`|(10-(-30))/sqrt(16+9)|=d`
`d=|(10+30)/5|`
`d=8`
`r=4`
O is mid point of AB
`(x,y)=((3-1)/2,(2-6)/2)=(1,-2)`
A circle passes through center`(1,-2)`
`(x-1)^2+(y+2)^2=4^2`.
Discussion
2.

If `y=2x`is a chord of the circle `x^2+y^2-10 x=0`, find the equation of a circle with this chord asdiameter.

Answer» `x^2+y^2-10x=0`
`x^2(2x)^2-10x=0`
`x^2+4x^2-10x=0`
`5x^2-10x=0`
`5x(x-2)=0`
`(x-x_1)(x-x_2)+(y-y_1)(y-y_1)=0`
`(x-0)(x-2)+(y-0)(y-4)=0`
`x(x-2)+y(y-4)=0`
`x^2+y^2-2x-4y=0`.
Discussion
3.

If `2x^2+lambdax y^+2y^2+(lambda-4)x+6y-5=0,`is the equation of a circle, then its radius is :

Answer» `2x^2+lambdaxy+2y^2+(lambda-4)x+6y-5=0`
`x^2+lambda/2xy+y^2+(lambda-4)/2x+3y-5/2=0`
`lambda/2=0,lambda=0`
`x^2+0+y^2-2x+3y-5/2=0`
`C(1,-3/2)`
`r=sqrt(1+5/4+5/2)`
`r=sqrt((4+9+10)/4)`
`r=sqrt23/2`.
Discussion
4.

Find the radius of the circle `(xcosalpha+ysinalpha-a)^2+(xsinalpha-ycosalpha-b)^2=k^2,ifalpha`varies, the locus of its centre is again a circle. Also,find its centre and radius.

Answer» `x&cos^2alpha+y^2sin^2alpha+a^2+2xysinalphacosalpha-2aysinalpha-2axcosalpha+x^2sin^2alpha+y^2cos^2alpha+b^2-2xysinalphacosalpha+2bycosalpha-2bxsinalpha=k^2`
`x^2+y^2-2x(acosalpha-bsinalpha)+2y(bcosalpha-asinalpha)+a^2+b^2-k^2=0`
`r=sqrt(g^2+f^2-c)`
`r=sqrt((acosalpha+bsinalpha)^2+(bcosalpha-asinalpha)^2-a^2-b^2+k^2)`
`r=sqrt(a^2+b^2+0-a^2-b^2+k^2`
`r=k`
`x=acosalpha+bsinalpha`
`y=-(asinalpha-bcosalpha)`
`x^2+y^2=a^2cos^2alpha+2ab sinalphacosalpha+b^2sin^2alpha+a^2sin^2alpha-2ab sinalphacosalpha+b^2cos^2alpha`
`x^2+y^2=a^2+b^2`
Circle r=`sqrt(a^2+b^2`
`C=(0,0)`.
Discussion