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# Geometry

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Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$

Note: The point $(x,y)$ lies in the first quadrant if both $x$ and $y$ are positive.

Apr 23, 2024

#1
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Construct a circle centered at the origin with a radius of 1

The equation of this  circle  is

x^2 + y^2   = 1         (1)

Let P, Q lie on this circle

Line  through OP  has the equation

y = 4x

Sub this into (1)  to find the x coordinate of P

x^2 + (4x)^2   =1

17x^2  = 1

x^2 = 1/17

x =1/sqrt 17

y = 4/sqrt 17

P = (1/sqrt 17, 4/sqrt 17)

Similarly, line through OQ  has the equation

y =5x

Sub this into (1) to find the x coordinate of Q

x^2 + (5x)^2  =1

26x^2  = 1

x^2 = 1/26

x =1/sqrt 26

y = 5/sqrt 26

Q = ( 1/sqrt 26,  5/sqrt 26)

Slope of  PQ =

[ 4/sqrt 17 - 5/sqrt 26 ] / [ 1/sqrt 17 - 1/sqrt 26 ]   =   [ 5 sqrt 26 - 20 ] / [sqrt 26 - 4 ]  =

[ 4sqrt 26 - 5sqrt 17 ] / [ sqrt 26 - sqrt 17 ]  =

[ 4sqrt 26 - 5sqrt 17 ] [ sqrt 26 + sqrt 17 ] / 9  =

[4 * 26  - sqrt (442) - 85 ]  / 9  =

[19 - sqrt 442 ]  /  9   ≈  -0.225

Apr 23, 2024