Question

Find the value of P if the line 3x + 4y – P = 0 is a tangent to the circle x² + y² = 16.

Answer 1

The condition for a line y = mx + c to be a tangent to the circle x² + y² = a² is c² = a² (1 + m²)

Equation of the line is 3x + 4y – P = 0

Equation of the circle is x² + y² = 16

4y = -3x + P

y = `(-3)/4x + "P"/4`

∴ m = `(-3)/4`, c = `"P"/4`

Equation of the circle is x² + y² = 16

∴ a² = 16

Condition for tangency we have c² = a²(1 + m²)

⇒ `("P"/4)^2 = 16 (1 + 9/16)`

⇒ `"P"^2/16 = 16(25/16)`

⇒ P² = 16 × 25

⇒ P = ± `sqrt16 xx sqrt25`

⇒ P = ±4 × 5

⇒ P = ±20