State whether the statements True or False. Justify

If the line lx + my = 1 is a tangent to the circle x² + y² = a², then the point (l, m) lies on a circle.

lx + my = 1

my = 1 – lx

Slope of tangent = -l

x² + y² = a²

(x – 0) ² + (y – 0) ² = a²

Centre = (0, 0) Radius = a units

Slope of line perpendicular to tangent= {Product of slopes of perpendicular lines = -1}

Equation of line perpendicular to tangent i.e.., Radius of the circle

ly = x – 0

Distance of centre (0,0) from the line lx + my = 1 is equal to the radius ‘a’.

Perpendicular Distance (Between a point and line) = , whereas the point is and the line is expressed as ax + by + c = 0

Squaring both the sides,

Hence, the locus of (l,m)is, which is a circle.

TRUE