Question

The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x − 4y = m at two distinct points if

a. −35 < m < 15

b. 15 < m < 65

c. 35 < m < 85

d. −85 < m < −35

Answer 1

We have,3x−4y=k⇒3x−4y−k=0x2+y2−4x−8y−5=0Center of circle x2+y2+2gx+2fy+c=0 is (−g,−f)Hence center of given circle=(2,4)Radius=g2+f2−c‾‾‾‾‾‾‾‾‾‾‾√=22+42−(−5)‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=4+16+5‾‾‾‾‾‾‾‾‾‾√=5 unitsIf line intersects circle at two points then length of perpendicular from center of circle must beless than radius of circle.⇒∣∣3.2−4.4−k∣∣32+42√<5⇒∣∣6−16−k∣∣5<5⇒∣∣−(k+10)∣∣<25⇒∣∣k+10∣∣<25if∣∣x∣∣<a, then−a<x<a⇒−25<k+10<25⇒−25−10<k+10−10<25−10⇒−35<k<15 (Answer)