Find the equations of circles with radius√13 units and touching 2x-3y+1=0 at (1,4)

Asked by kalivelapriya18 | 6th Apr, 2022, 03:14: PM

Expert Answer:

As the two circles touches the same line at the same point, it becomes common tangent to the circles.
Let O and O' be the centres of the circles.
Let the equation of line OO' be 3x + 2y + k = 0
Since OO' passes through (1, 1), we get
k = -5
So, the equation of line OO' is 3x + 2y - 5 = 0
The centres A and B are given by
open parentheses 1 plus-or-minus square root of 13 open parentheses negative fraction numerator 2 over denominator square root of 13 end fraction close parentheses comma space 1 plus-or-minus square root of 13 open parentheses fraction numerator 3 over denominator square root of 13 end fraction close parentheses close parentheses
equals open parentheses 1 minus 2 comma space 1 plus 3 close parentheses space space or space left parenthesis 1 plus 2 comma space 1 minus 3 right parenthesis
equals left parenthesis negative 1 comma space 4 right parenthesis space or space left parenthesis 3 comma space minus 2 right parenthesis
So comma space the space equations space of space the space circles space are space
left parenthesis straight x plus 1 right parenthesis squared plus left parenthesis straight y minus 4 right parenthesis squared equals 13 space left parenthesis straight x minus 3 right parenthesis squared plus left parenthesis straight y plus 2 right parenthesis squared equals 13

Answered by Renu Varma | 8th Apr, 2022, 11:14: PM

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