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Question

If the curves, x2-6x+y2+8=0 and x2-8y+y2+16-k=0,(k>0) touch each other at a point, then the largest value of k is:


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Solution

Step 1: Determine the center & radius

Given, curves, x2-6x+y2+8=0 and x2-8y+y2+16-k=0,(k>0) touch each other at a point.

Comparing curve x2-6x+y2+8=0 with x2+y2+2gx+2fy+c=0 where -g,-f is center & radius=f2+g2-c we have:

Center≡3,0

radius=32+0-8=1

Comparing curve x2-8y+y2+16-k=0,(k>0) with x2+y2+2gx+2fy+c=0 we get:

Center≡0,4

radius=0+42+k=16+k

Step 2: Find distance between center

Circle touch each other if C1C2=r1±r2
Distance between center C2(0,4) and C1(3,0) is either k+1 or |k-1|.

C1C2=32+42=5

Step 3: Find maximum value of k

To find the maximum value of kput C1C2=r1±r2.

⇒k+1=5⇒k=4⇒k=16

For, |k-1|=5
⇒k-1=5⇒k=5+1⇒k=6⇒k=36
Maximum at k=36.
Therefore, the largest value of k is 36.


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