Question

Find the locus of the foot of the perpendicular from the origin to a variable straight line which always
passes through a fixed point (a, b).​

Answer 1

Given:

Variable straight line which always  passes through a fixed point (a, b).​

To Find:

The locus of the foot of the perpendicular from the origin to the variable straight line

Solution:

Let  A ( p , q ) be the foot of the perpendicular from the origin to the line.

Then line joining origin and the A ( p,q ) will be perpendicular to the line.

• Slope of line x slope of OA = -1

Now,

• Slope of the line = (q - b)/( p - a)

• Slope of OA = (q - 0 )/ (p - 0)

Applying the condition,

• (q - b )/( p -a ) x  q / p  = -1
• q² - qb = - 1 ( p² -pa )
• p² - pa + q² -pb = 0
• ( p , q ) can vary as ( x, y )

Therefore locus of the  foot of the perpendicular from the origin to the variable straight line x² + y² -ax - by = 0