# If the straight lines #ax + by + c = 0# and #x cos(alpha) + y sin(alpha) = c# enclose an angle #pi/4# between them and meet the straight line #x sin(alpha) - y cos(alpha) = 0 # in the same point between them, then?

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A) #a^2 + b^2 = c^2#

B) #a^2 + b^2 = 2#

C) #a^2 + b^2 = 2c^2#

D)# a^2 + b^2 = 4#

A)

B)

C)

D)

##### 3 Answers

#### Explanation:

From the lines

we have

but from

the intersection point gives us

and after substituting into

then

Given lines are

As per given condition above three given lines pass through a fixed point.

Hence

Now equations of the angle bisectors between

Bisector -1

Bisector 2

Comparing

So

Comparing

So

B)

#### Explanation:

An alternative quick and dirty method to decide which of the given options is correct is to consider a particular example...

Let

Then the third straight line is:

#xsqrt(2)/2-ysqrt(2)/2 = 0#

which simplifies to

The second straight line is:

#xsqrt(2)/2+ysqrt(2)/2 = 3sqrt(2)#

which simplifies to

These intersect at

The first straight line passes through this intersection point and is horizontal or vertical. That is:

#0x-sqrt(2)y+3sqrt(2) = 0#

or:

#-sqrt(2)x+0y+3sqrt(2) = 0#

In either case