The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at

1.	The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if … (a) 15 < m < 65(b) 35 < m < 85 (c) -85 < m < -35 (d) -35 < m < 15
Answer» (d) -35 < m < 15 x² + y² – 4x – 8y – 5 = 0 3x – 4y = m Solving (1) and (2) we get from (2) ⇒ 3x = 4y + m x = (4y + m)/3 Substituting x value in (1) we get (4y + m/3)² + y² - 4(4y + m/3) - 8y - 5 = 0 It is a quadratic in y and given that the roots are distinct ⇒ b² – 4ac > 0 On simplifying we get ⇒ – 9m² – 18w + 4725 > 0 ⇒ m² + 20m – 525 < 0 ⇒ (m + 35) (m – 15) ≤ 0 ⇒ m lies between -35 and 15 ie„ – 35 < m < 15

The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if … (a) 15 < m < 65(b) 35 < m < 85 (c) -85 < m < -35 (d) -35 < m < 15

Answer»

(d) -35 < m < 15

x² + y² – 4x – 8y – 5 = 0

3x – 4y = m Solving (1) and (2) we get

from (2) ⇒ 3x = 4y + m

x = (4y + m)/3

Substituting x value in (1) we get

(4y + m/3)² + y² - 4(4y + m/3) - 8y - 5 = 0

It is a quadratic in y and given that the roots are distinct

⇒ b² – 4ac > 0

On simplifying we get

⇒ – 9m² – 18w + 4725 > 0

⇒ m² + 20m – 525 < 0

⇒ (m + 35) (m – 15) ≤ 0

⇒ m lies between -35 and 15 ie„ – 35 < m < 15