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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:

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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A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of identical shape and material are now added to P as shown in the figure. The ray will suffer

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A bob of mass $M$ is suspended by a massless string of length $L$ . The horizontal velocity $V$ at position $A$ is just sufficient to make it reach the point $B$ . The angle $theta$ at which the speed of the bob is half of that at $A$ , satisfies

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V to the power of 2 end exponent equals U to the power of 2 end exponent minus 2 g left parenthesis L minus L cos invisible function application theta right parenthesis

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open parentheses fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator negative 1 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction close parentheses

open parentheses fraction numerator 2 over denominator 3 end fraction comma fraction numerator negative 1 over denominator 3 end fraction comma fraction numerator 2 over denominator 3 end fraction close parentheses

The foot of the perpendicular on the line $3 x plus y equals lambda$ drown from the origin is C if the line cuts the x-axis and y-axis at A and B respectively then BC : CA is

Let direction ratios of the required line be <a, b, c>
Therefore a - 2 b - 2 c = 0
And 2 b + c = 0
Þ c = - 2 b
a - 2 b + 4b = 0 Þ a = - 2 b
Therefore direction ratios of the required line are <- 2b, b, - 2b> = <2, - 1, 2>
direction cosines of the required line

open parentheses fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator negative 1 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction comma fraction numerator 2 over denominator square root of 2 to the power of 2 end exponent plus 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end root end fraction close parentheses

open parentheses fraction numerator 2 over denominator 3 end fraction comma fraction numerator negative 1 over denominator 3 end fraction comma fraction numerator 2 over denominator 3 end fraction close parentheses

parallel

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= –3
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fraction numerator z subscript 1 end subscript plus 1 over denominator 2 end fraction

= –3
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. Hence (c) is the correct choice.

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\frac{l}{- 1} = \frac{m}{+ 1} = \frac{n}{3}

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. Hence (c) is the correct choice.

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Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0 but normal to the plane will be perpendicular to x-axis. Hence (c) is the correct answer.

The value of $integral subscript 0 superscript 100 left curly bracket square root of x right curly bracket d x$ (where {x} is the fractional part of x) is

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parallel

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