Question

 Find the equation of the circle having the pair of lines x² + 2xy + 3x + 6y = 0 as its normals and having the size just sufficient to contain the circle x(x − 4) + y(y − 3) = 0.

Answer 1

We have

The point of intersection of the lines =(−3, 3/2), which is the centre of the required circle. Also the centre and radius of the circle x(x − 4) + y(y − 3) = 1 are (2, 3/2) and 5/2, respectively. If the required circle is to just contain the circle with centre (2, 3/2) and radius 5/2, then this circle should touch the required circle internally. If r is the radius of the required circle the distance between their centres (−3, 3/2) and (2, 3/2) must be equal to r − 5/2 (since r > 5/2). Therefore

 Hence, the equation of the required circle is