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My book says -

The equation of the family of circles touching the circle $S = 0$ and the line $L = 0$ at their point of contact $P$ is -
                                                                   $S + \lambda L = 0$     where $\lambda $ is a parameter.

In the above equation, $S=0$ and $L=0$ both satisfy the coordinates of $P$. Hence, $S + \lambda L = 0$ also satisfies the coordinates of $P$. I am not able to find any condition that makes the circle $S + \lambda L = 0$ only touch the circle $S=0$ and the line $L=0$.

I feel this is the equation of family of circles passing through point of contact $P$ rather than the equation of family of circles touching the circle and the line at $P$ exclusively.

What am I missing here?

DoubtExpert
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2 Answers2

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if $(S+ \lambda L = 0)$ intersected the line or the circle at any other point, it would also intersect the other object there, so the initial circle and line would have two intersection points. Since this is not the case, $P$ is the only intersection of $(S+ \lambda L=0)$ with either the circle or the line, so it is tangent to them at $P$.

mercio
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  • Oh...a circle $S + \lambda L = 0$ cannot exclusively intersect the line $L=0$ or $S=0$ on passing through their point of contact. That how I considered your answer ought to be right. Thanks! – DoubtExpert Sep 30 '17 at 07:50
  • You can also look at the gradient of the two equations at $P$, since they have the same direction, any linear combination of them will have that same direction, so the slope of the new circle is the still the same as the slope of the line and the original circle. – mercio Sep 30 '17 at 07:56
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Let $C = S+\lambda L$

$C = 0$ is a second degree equation in both $x$ and $y$ and since it is tangent to both $S = 0$ and $L = 0$, it has two coincident roots with both of the curves. Note that $C=0$ cannot intersect either of the two again as second degree equation cannot have more than two roots!

jonsno
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  • How do you know $C=0$ is tangent to the other two curves? That basically was my question. – DoubtExpert Sep 30 '17 at 07:54
  • what does touching mean. – jonsno Sep 30 '17 at 07:55
  • That is what we are trying to find right - whether it is touching or not? – DoubtExpert Sep 30 '17 at 07:58
  • Dear friend, $C = 0$ passes through intersection of $S=0$ and $L=0$. In general, there are two separate points of intersection of $S = 0$ and $L = 0$. In this case $L = 0$ is a chord to both $C = 0$ and $S = 0$. But in question you say its only touching them. Then you need $L = 0$ to be tangent to $S = 0$, and then by argument in my answer, $C =0$ will also only touch them, not intersect them! – jonsno Sep 30 '17 at 08:03
  • I don't know understand the lead to your answer. There is already a circle $S=0$ and a line $L=0$ existing on the plane and also $L=0$ is tangent to $S=0$. We need to find another circle which is tangent to both of these curves. The $C=0$ which you have mentioned, does not signify that it touching the curves but only tells that it passes through $P$. So, I was asking how does $C=0$ also signify that it is tangent to both the curves. But you start your answer itself by assuming that $C=0$ is tangent. Is that right? – DoubtExpert Sep 30 '17 at 08:23
  • Let us continue this discussion in chat. – DoubtExpert Sep 30 '17 at 08:23