Show that 2x + y + 6 = 0 is a tangent to x2 + y2 + 2x – 2y – 3 = 0

1.	Show that 2x + y + 6 = 0 is a tangent to x2 + y2 + 2x – 2y – 3 = 0. Find its point of contact.
Answer» 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 x + (-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)

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

Answer»

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

x + (-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)

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

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