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If the curves, x^{2}-6x+y^2+8=0\: and\: x^2-8y+y^2+16-k=0,(k>0) touch each other at a point, then the largest value of k is__________. 

 

Option: 1

15


Option: 2

30


Option: 3

51


Option: 4

36


Answers (1)

best_answer

 

 

Intersection of Two Circle -

CASE 2

When two circles touch each other externally
\\\mathbf{\left|C_{1} C_{2}\right|=r_{1}+r_{2}},

i.e, the distance between the centres is equal to the sum of radii, then two circles touch externally.

In this case, two direct common tangents are real and distinct while the transverse tangents are coincident.

In this case, point of contact P divides C1 and C2 internally in the ratio r1:r2.
\\\mathrm{\frac{C_1P}{C_2P}=\frac{r_1}{r_2}}\\\\\mathrm{Coordinate\;of\;point \;P\;is\;\;\left(\frac{r_{1} x_{2}+r_{2} x_{1}}{r_{1}+r_{2}}, \frac{r_{1} y_{2}+r_{2} y_{1}}{r_{1}+r_{2}}\right)}\\\\\begin{array}{c}{\text { The equation of tangent at point } P \text { is } S_{1}-S_{2}=0, \text { where }} \\ {S_{1}=0 \text { and } S_{2}=0 \text { are equations of circles. }}\end{array}

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Two circles touch each other if \mathrm{C}_{1} \mathrm{C}_{2}=\left|\mathrm{r}_{1} \pm \mathrm{r}_{2}\right|

Distance between C2(3, 0) and C1(0, 4) is either \sqrt{k}+1 \text { or }|\sqrt{k}-1|

Also \mathrm{C}_{1} \mathrm{C}_{2}=\sqrt{4^2+3^2}=5

\Rightarrow \sqrt{k}+1=5 \text { or }|\sqrt{k}-1|=5 \quad \Rightarrow k=16 \text { or } k=36

Maximum value of K is 36

Correct Option 4

Posted by

Deependra Verma

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