Let #l# be a line described by equation ax+by+c=0 and let #P(x,y)# be a point not on #l#. Express the distance, #d# between #l and P# in terms of the coefficients #a, b and c# of the equation of line?

Answer 1

#d = (c + a x_0 + b y_0)/sqrt(a^2 + b^2)#

Let #l->a x + b y + c=0# and #p_0 = (x_0,y_0)# a point not on #l#.
Supposing that #b ne 0# and calling #d^2=(x-x_0)^2+(y-y_0)^2# after substituting #y=-(a x+c)/b# into #d^2# we have
#d^2=(x - x_0)^2 + ((c + a x)/b + y_0)^2#. The next step is find the #d^2# minimum regarding #x# so we will find #x# such that
#d/(dx)(d^2) = 2 (x - x_0) - (2 a ((c + a x)/b + y_0))/b = 0#. This occours for
#x = (b^2 x_0 - a b y_0-a c)/(a^2 + b^2)# Now, substituting this value into #d^2# we obtain
#d^2=(c + a x_0 + b y_0)^2/(a^2 + b^2)# so
#d = (c + a x_0 + b y_0)/sqrt(a^2 + b^2)#
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Answer 2

The distance ( d ) between the line ( l ) described by the equation ( ax + by + c = 0 ) and a point ( P(x, y) ) not on ( l ) can be expressed using the formula:

[ d = \frac{|ax + by + c|}{\sqrt{a^2 + b^2}} ]

Where:

  • ( a ), ( b ), and ( c ) are the coefficients of the equation of the line ( l ).
  • ( x ) and ( y ) are the coordinates of the point ( P(x, y) ).
  • ( |ax + by + c| ) represents the absolute value of the expression ( ax + by + c ).
  • ( \sqrt{a^2 + b^2} ) is the square root of the sum of squares of the coefficients ( a ) and ( b ).
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Answer 3

The distance (d) between a line described by the equation (ax + by + c = 0) and a point (P(x, y)) not on the line can be expressed as:

[ d = \frac{|ax + by + c|}{\sqrt{a^2 + b^2}} ]

So, the distance (d) is given by the absolute value of the expression (ax + by + c) divided by the square root of the sum of the squares of coefficients (a) and (b).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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