Question

A variable circle is drawn to touch the line 3x-4y=10 and also circle x^2+y^2=4 externally then

Answer 1

4h + 3k = 0

Step-by-step explanation:

x² + y² = 4

=> (x - 0)² +(y -0)² = 2²

Circle center (0,0)  & radius = 2

Now let say variable circle

(x - h)² + (y - k)² = r²

=> ( h - 0)² + (k - 0)² = (r + 2)²

=> h² + k² = (r + 2)²

3x-4y=10

r = | (3h - 4k - 10 )/ √(3² + 4²)

=> r = | (3h - 4k - 10 )/5 |

h² + k² =  ((3h - 4k - 10 )/5  + 2 )²

=> 25h² + 25k²  = (3h - 4k)²

=> 25h² + 25k² = 9h² + 16k² - 24hk

=> 16h² + 9k²  + 24hk = 0

=> (4h + 3k)² = 0

=> 4h + 3k = 0

Hence variable circle equation

4h + 3k = 0

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