There are two circles whose equations are S 1- x 2- y 2-49 and S 2- x 2- y 2-16 x -12 y - p 2-0- p - Z- If these circles haveexactly two common tangents- then the sum of all the possible values of p is

For circle S_2=x^2+y^2+16x+12y+p^2=0, the centre is ( 8, 6) and radius is √(( 8)^2+( 6)^2 p^2) Then nbsp;√(( 8)^2+( 6)^2 p^2)0 10≤ p≤10 The circles should in ...

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Standard XII

Mathematics

Radical Axis

There are two...

Question

There are two circles whose equations are $S_{1} = x^{2} + y^{2} = 49$ and $S_{2} = x^{2} + y^{2} + 16 x + 12 y + p^{2} = 0, p \in Z$ . If these circles have exactly two common tangents, then the sum of all the possible values of $p$ is

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Solution

For circle $S_{2} = x^{2} + y^{2} + 16 x + 12 y + p^{2} = 0$ , the centre is $(- 8, - 6)$ and radius is $\sqrt{(- 8)^{2} + (- 6)^{2} - p^{2}}$
Then $\sqrt{(- 8)^{2} + (- 6)^{2} - p^{2}} > 0$
$- 10 \leq p \leq 10$
The circles should intersect each other at two points to have exactly two common tangents. So,
$r_{1} + r_{2} > d > | r_{2} - r_{1} |$
$d = \sqrt{(- 8 - 0)^{2} + (- 6 - 0)^{2}} = 10$
$r_{1} + r_{2} > d \Rightarrow 7 + \sqrt{100 - p^{2}} > 10$
$\Rightarrow \sqrt{100 - p^{2}} > 3$
$\Rightarrow 100 - p^{2} > 9$
$\Rightarrow - \sqrt{91} \leq p \leq \sqrt{91}$
Also $d > | r_{2} - r_{1} |$
$10 > 7 - \sqrt{100 - p^{2}}$ or
$10 > \sqrt{100 - p^{2}} - 7$
if $10 > 7 - \sqrt{100 - p^{2}}$
$\Rightarrow \sqrt{100 - p^{2}} > - 3$ which is always true
If $10 > \sqrt{100 - p^{2}} - 7$
$p^{2} > - 189$ which will be also always true
so $- \sqrt{91} \leq p \leq \sqrt{91}$
Now $p \in Z$ so $p = - 9, - 8, - 7, . . . . . . .8, 9$
$\Rightarrow$ sum of all values $p = 0$

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There are two circles whose equations are S1=x2+y2=49 and S2=x2+y2+16x+12y+p2=0, p∈Z. If these circles have exactly two common tangents, then the sum of all the possible values of p is

There are two circles whose equations are $S_{1} = x^{2} + y^{2} = 49$ and $S_{2} = x^{2} + y^{2} + 16 x + 12 y + p^{2} = 0, p \in Z$ . If these circles have exactly two common tangents, then the sum of all the possible values of $p$ is