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