the locus of the point of intersection of tangents of a square divided by 2 square + 5 square by metre is equal to one
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Locus of point of intersection of tangents tothe circle x² + y2 = a which makescomplementary angles with X-axis is1) x² y² = 0 2) x² + y² = 03) xy = 0 4) x² + y2 = 2a
i.e.,y=mx\pm r\sqrt{1+m^{2}}
y-mx=\pm r\sqrt{1+m^{2}}
Squaring on both sides, we get
(y-mx)^2=(\pm r\sqrt{1+m^{2}})^2
y^2+m^2x^2-2mxy=r^2(1+m^2)
y^2+m^2x^2-2mxy=r^2+r^2m^2
m^2(x^2-r^2)-2mxy+y^2-r^2=0
For the lines to make complemetary angles with the x-axis the product of the roots of above equation is equal to -1
i.e.,,m1m2 = -1
\Rightarrow m_1m_2=(y^2-r^2)/(x^2-r^2)=-1
On simplifying you get the equation of a circle known as director circle
x2+y2=2a2 [since here r=a]
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