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.
The equation for a line is y(x)=ax+b.Because it intersects the y axis at x=0 at y=1y(x)=b=1a is equal to the gradient of the function so the line is given as y(x)=3x+1In order to get the two intersections we need to solve the following equation system:(1) x2+y2=1(2) y=3x+1
(1) x2+(3x+1)2=1 x2+9x2 +1+6x=1 10x2+6x=0Then solving it for x we get x=0 y=1 and x=-3/5 y=-4/5
CB
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l1 and l2 are tangents of a circle. l1 intersects the circle at (3-√3,5) with a gradient of √3, and l2 intersects the circle at (3+√2,4+√2) with a gradient of -1. Find the centre of the circle, and hence find the radius of the circle.