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Maths-

General

Easy

Question

The equation of circle which touches the line 5x + 12 y =1 and which has its centre at (3, 4) is:

$(x - 3)^{2} + (y - 4)^{2} = {(\frac{62}{11})}^{2}$
$(x - 3)^{2} + (y - 4)^{2} = {(\frac{62}{17})}^{2}$
$(x - 3)^{2} + (y - 4)^{2} = {(\frac{62}{13})}^{2}$
none of these

hint Hint:

Find the radius of the circle and substitute required values in the standard equation of the circle.

The correct answer is: $left parenthesis x minus 3 right parenthesis to the power of 2 end exponent plus left parenthesis y minus 4 right parenthesis to the power of 2 end exponent equals open parentheses fraction numerator 62 over denominator 13 end fraction close parentheses to the power of 2 end exponent$

$G i v e n comma C e n t r e space o f space t h e space c i r c l e space open parentheses x subscript 1 comma y subscript 1 close parentheses equals open parentheses 3 comma 4 close parentheses L i n e space a x plus b y plus c equals 0 space rightwards double arrow 5 x plus 12 y equals 1 T o space f i n d comma E q u a t i o n space o f space t h e space c i r c l e S o l u t i o n comma W e space k n o w space t h a t space t h e space p e r p e n d i c u l a r space d i s tan c e space f r o m space t h e space p o i n t space t o space t h e space l i n e space i s fraction numerator open vertical bar a x subscript 1 plus b y subscript 1 plus c close vertical bar over denominator square root of a squared plus b squared end root end fraction L e t space u s space a s s u m e space r space t o space b e space t h e space r a d i u s space o f space t h e space c i r c l e. space H e n c e comma r equals fraction numerator open vertical bar a x subscript 1 plus b y subscript 1 plus c close vertical bar over denominator square root of a squared plus b squared end root end fraction equals fraction numerator open vertical bar 5 cross times 3 plus 12 cross times 4 plus open parentheses negative 1 close parentheses close vertical bar over denominator square root of 5 squared plus 12 squared end root end fraction equals fraction numerator open vertical bar 15 plus 48 minus 1 close vertical bar over denominator square root of 25 plus 144 end root end fraction equals 62 over 13 N o w comma E q u a t i o n space o f space t h e space c i r c l e space i s open parentheses x minus x subscript 1 close parentheses squared plus open parentheses y minus y subscript 1 close parentheses squared equals r squared open parentheses x minus 3 close parentheses squared plus open parentheses y minus 4 close parentheses squared equals open parentheses 62 over 13 close parentheses squared space$

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