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The circle c has equation  x^2 + y^2 = 1. The line l has gradient 3 and intercepts the y axis at the point (0, 1). c and l intersect at two points. Find the co-ordinates of these points.

 Jul 27, 2017
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If the line's slope is  3  and y-intercept is  1 , the slope-intercept form of the line is:

y  =  3x + 1

 

And the equation for the circle is:

x2 + y2  =  1                        We want to find what x is when y = 3x + 1. So substitute 3x + 1 in for y.

x2 + (3x + 1)2  =  1

x2 + (3x +1)(3x + 1)  =  1

x2 + 9x2 + 6x + 1  =  1

10x2 + 6x  =  0                   Factor out an  x  from both terms.

x(10x + 6)  =  0                   Set each factor equal to zero and solve for  x .

 

x  =  0       or       x  =  -3/5

 

Now we can plug these  x  values into the equation of the line to find the  y  coordinate of the intersection points.

 

When  x  =  0 ,                       When  x  =  -3/5 ,

y  =  3(0) + 1                           y  =  3(-3/5) + 1

y  =  1                                     y  =  -4/5

 

So...the coordinates of the two points are:    (0, 1)   and   (-3/5, -4/5)

 

Here's a graph: https://www.desmos.com/calculator/0x0hcauvxk

 Jul 27, 2017

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