+0

Coordinate geo

0
5
2
+19

Let O be the origin. Points P and Q lie in the first quadrant. The slope of line segment OP is 1 and the slope of line OQ segment is 3. If OP=OQ then compute the slope of line segment PQ.

Jun 21, 2024

#1
+10
0

$$Q$$ lies on the line $$y=3x$$
$$P$$ lies on the line $$y=x$$

WLOG lets $$Q=(1,3)$$, therefore $$OP=OQ=\sqrt{10}$$

Thus we can say $$P=(\sqrt{5},\sqrt5)$$

Slope between 2 points is $$s=\large{\frac{y_2-y_2}{x_2-x_1}=\frac{\sqrt5-3}{\sqrt5-1}=\boxed{\frac{1-\sqrt5}{2}}}$$

Jun 21, 2024
#2
+129725
+1

Let  P , Q  lie on a unit circle

This circle has the equation x^2 + y^2  =1

Equation of line containg OP  is  y =x

Equation of line  containing OQ  is  y = 3x

The x coordinate of P  can be  found   as

x^2 + x^2 = 1

2x^2 =1

x^2  =1/2

x = 1/sqrt 2      and  y =1/sqrt 2

The x coordinate of Q  can be found as

x^2 + (3x)^2  = 1

10x^2 = 1

x^2  =1/10

x= 1/sqrt 10     y = 3/sqrt 10

Slope  between   PQ   is

[ 1/sqrt 2 - 3/sqrt 10 ] / [ 1/sqrt 2 - 1/sqrt 10 ]  =

[ sqrt 10 - 3sqrt 2 ] / [sqrt 10 - sqrt 2]  =

[ sqrt 10 -3sqrt 2 ] [ sqrt 10 + sqrt 2 ] / [ 10 -2]  =

[ 10 -2sqrt 20 - 6 ] / [ 8]  =

[ 4 - 4sqrt 5 ] / 8  =

[ 1 - sqrt 5 ] / 2

Jun 22, 2024