Question

Answer the following :

Show that 2x + y + 6 = 0 is a tangent to x² + y² + 2x – 2y – 3 = 0. Find its point of contact

Answer 1

Given equation of circle is

x² + y² + 2x – 2y – 3 = 0   ........(i)

Given equation of line is

2x + y + 6 = 0

∴ y = – 6 – 2x  ........(ii)

Substituting y = – 6 – 2x in (i), we get

x² + (– 6 – 2x)² + 2x – 2(– 6 – 2x) – 3 = 0

∴ x² + 36 +24x + 4x² + 2x + 12 + 4x – 3 = 0

∴ 5x² + 30x + 45 = 0

∴ x² + 6x + 9 = 0

∴ (x + 3)² = 0

∴ x = – 3

Since, the roots are equal.

∴ 2x + y + 6 = 0 is a tangent to

x² + y² + 2x – 2y – 3 = 0

Substituting x = – 3 in (ii), we get

y = – 6 – 2(– 3) = – 6 + 6 = 0

∴ Point of contact = (– 3, 0)