show that, whatever be the value of α, the lines x cos α + y sin α = a and - Maths - Conic Sections

Question

show that, whatever be the value of α, the lines x cos
α + y sin α = a and x sin α-ycosα=a are
tangents to the circle x²+y²=a²

Anuradha Sharma · Accepted Answer

The parametric point on circle x2+y2=a2 is given as, acosα, asinα, so equation of tangent at this point is, xx'+yy'=a2xacosα+yasinα=a2or xcosα+ysinα=a
1 is a tangent and that is what we have to show
Now another point β on circle with parametric difference 3π2 can be given asPutting this in equation 1 we get,  xcos3π2+α+ysin3π2+α=axsinα-ycosα=aSo this is also an tangent

show that, whatever be the value of α, the lines x cos α + y sin α = a and x sin α-ycosα=a are tangents to the circle x²+y²=a²