Below, I have posted a question and a solution I was provided. The problem is that I don't understand how the part bordered by AAA was arrived at. Specifically , & why it's simply multiplied against ? After multiplying that expression, I end up with , & not . Can anyone help clarify this for me, or is there a better approach?
Q. is the equation of a circle. 2 lines, L & M, intersect at (2, -2). The distance from the centre of the circle to each line is . Find the equation of L & M.
Attempt: Line equation for (2,-2):
From circle equation: centre c = (-1, 2)
Sub c into the line equation:
The distance from the point (-1, 2) to this line is
Ans. (From text book): &
I don't think that this is necessarily a better approach, but it is different.
The circle has a centre at and has a radius of 5. If we draw a second smaller circle with the same centre and with radius then what we are looking for are the two tangents from the point (which incidently lies on the larger circle), to the smaller one.
The smaller circle will have equation or,
and a tangent to this circle at the point will have equation
This, (these), has (have) to pass through the point so
The point lies on the smaller circle, so substituting allows us to find the co-ordinates of the two points of contact. These turn out to be
Corresponding co-ordinates are
The equations of the two tangents can then be calculated (and agree with the given answers).