if the line’s 3x-4y+4= and 6x-8y+7= are tangents to a circle then find the radius of the circle About the author Arya
Answer: Equation of the given tangents are- t 1 :3x−4y+4=0 t 2 :6x−8y−7=0⇒3x−4y− 2 7 =0 Here, a=3,b=−4,c 1 =4,c 2 =− 2 7 Since slopes of the given tangents are equal, i.e., 4 3 ∴ the given tangents are parallel. As we know that, the distance between two parallel lines given by, d= a 2 +b 2 ∣c 2 −c 1 ∣ ∴ distance between t 1 &t 2 = 3 2 +(−4) 2 ∣ ∣ ∣ ∣ ∣ ∣ 4−(− 2 7 ) ∣ ∣ ∣ ∣ ∣ ∣ = 9+16 ( 2 15 ) = 2×5 15 = 2 3 As we know that distance between two parallel tangents of a circle is equal to the diameter of that circle. ∴ diameter of given circle = 2 3 ∴ Radius of the given circle = 2 diameter = 2 ( 2 3 ) = 4 3 Hence, the radius of the given circle is 4 3 . Reply
Answer:
Equation of the given tangents are-
t
1
:3x−4y+4=0
t
2
:6x−8y−7=0⇒3x−4y−
2
7
=0
Here,
a=3,b=−4,c
1
=4,c
2
=−
2
7
Since slopes of the given tangents are equal, i.e.,
4
3
∴ the given tangents are parallel.
As we know that, the distance between two parallel lines given by,
d=
a
2
+b
2
∣c
2
−c
1
∣
∴ distance between t
1
&t
2
=
3
2
+(−4)
2
∣
∣
∣
∣
∣
∣
4−(−
2
7
)
∣
∣
∣
∣
∣
∣
=
9+16
(
2
15
)
=
2×5
15
=
2
3
As we know that distance between two parallel tangents of a circle is equal to the diameter of that circle.
∴ diameter of given circle =
2
3
∴ Radius of the given circle =
2
diameter
=
2
(
2
3
)
=
4
3
Hence, the radius of the given circle is
4
3
.