Consider a line \(L:3x+4y-10=0\).

Consider a circle \(S_1:x^2+y^2-16x-18y+109=0\).

Find the equation of a certain circle \(S_2\) which satisfies the 3 conditions below:

\(S_2\) touches \(S_1\) externally.

\(S_2\) touches \(L\).

Line joining the centers of \(S_1\) and \(S_2\) is perpendicular to \(L\).

Do post nice solutions and hints!

## Comments

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TopNewestThe equation of S1 can be rewritten as \[(x-8)^{2}+(y-9)^{2}=(6)^{2}\]

The distance of the line L from center of S1 is

\[\frac {3 (8)+4 (9)-10}{\sqrt {3^{2}+4^2}}=10\]

And since the radius of S1 is 6 The minimum distance between the S1 and the line is 4 which can also be seen as the diameter of the touching circle S2. Therefore the radius of S2 is 2. Now we have to find its centre.

The slope of the line perpendicular to L which is also the line joining their centers is 4/3.

Now using the parametric form

\[\frac {x-8}{3/5}=\frac {y-9}{4/5}=-8\]

Since the distance between the two centers is (6+2)=8 and - sign can be clear from the diagram.

x=16/5 y=13/5 the coordinates of the center of S2.

Equation is

\[(x-16/5)^{2}+(y-13/5)^{2}=(2)^{2}.\]

I might have done several mistakes please notify me if you find any. Hope I am right? – Satvik Choudhary · 2 years, 3 months ago

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The diagram – Satvik Choudhary · 2 years, 3 months ago

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Alternate way : Find intersection of \(L\) and the line joining centers. And use section formula with ratio \(4:1\) to obtain center.I obtained radius in the same way as yours.

Thanks @satvik choudhary ! – Nihar Mahajan · 2 years, 3 months ago

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– Satvik Choudhary · 2 years, 3 months ago

How to upload images i was just able to provide a link to it.Log in to reply

Don't type: no space , title , link – Nihar Mahajan · 2 years, 3 months ago

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The method i suggest is that, the common tangent of the circles touching externally is the radical axis.

let \(S_{2}=x^2+y^2+2gx+2fy+c=0\)

Given \(S_{1}=x^2+y^2-16x-18y+109=0\)

Radical axis of these circle is \(L\)

\(S_{1}-S_{2}=L\)

compare the coefficient of \(x\) and \(y\) and constant.

Hope it helps – Tanishq Varshney · 2 years, 3 months ago

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– Nihar Mahajan · 2 years, 3 months ago

I think satvik's way is easier. While solving by your method , I encountered difficulties in computing \(c\) which gave rise to a huge quadratic and I was f**ked! Anyway , I appreciate this idea of radical axis.Log in to reply

– Nihar Mahajan · 2 years, 3 months ago

Sorry to say but it is not necessary that \(S_1\) touches \(L\). :(Log in to reply

– Tanishq Varshney · 2 years, 3 months ago

but radical axis is a line perpendicular to the line joining their centresLog in to reply

– Nihar Mahajan · 2 years, 3 months ago

Oh! I get it. So the strategy is to construct a radical axis , and since the slopes are equal , we can get the radius of required circle by comparing the equation of the lines.Log in to reply

– Tanishq Varshney · 2 years, 3 months ago

actually the conditions u have put up are satisfied by the radical axis of circlesLog in to reply

– Satvik Choudhary · 2 years, 3 months ago

Actually if you calculate out then it will come out that L does not touch S1 so it can not be the radical axis for the two touching circles.Log in to reply

– Tanishq Varshney · 2 years, 3 months ago

Can u make a figure to justify ur statement, well i will post mine in few minutesLog in to reply

– Nihar Mahajan · 2 years, 3 months ago

I had the same doubt. Tanishq bhayya wants to say that if we "construct" the radical axis and compare it with \(L\) we get the requirements for equation of required circle.Log in to reply

– Satvik Choudhary · 2 years, 3 months ago

I tried to post my solution. I am having problem with the image uploading.Log in to reply

– Nihar Mahajan · 2 years, 3 months ago

What problem are you having with uploading the pic?Log in to reply

@Otto Bretscher @Tanishq Varshney @Chew-Seong Cheong @Jon Haussmann Please post hints/solutions. – Nihar Mahajan · 2 years, 3 months ago

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