1
:x
2
+y
2
−6x−9y+13=0
(x−3)
2
+(y−
2
9
)
2
−9−
4
81
+13=0
(x−3)
2
+(y−
2
9
)
2
=
4
65
Here,
r
1
=
2
65
C
1
=(3,
2
9
)
Equation of another circle-
S
2
:x
2
+y
2
−2x−16y=0
(x−1)
2
+(y−8)
2
−1−64=0
(x−1)
2
+(y−8)
2
=65
Here,
r
2
=
65
C
2
=(1,8)
Distance between the centre of two circles-
C
1
C
2
=
(3−1)
2
+(8−
2
9
)
2
C
1
C
2
=
4+
4
49
=
2
65
∣r
2
−r
1
∣=
∣
∣
∣
∣
∣
∣
65
−
2
65
∣
∣
∣
∣
∣
∣
=
2
65
∵C
1
C
2
=∣r
1
−r
2
∣
Thus the two circles touches each other internally.
Since the circle touches each other internally. The point of contact P divides C
1
C
2
externally in the ratio r
1
:r
2
, i.e.,
2
65
:
65
=1:2
Therefore, coordinates of P are-
⎝
⎜
⎜
⎜
⎜
⎜
⎛
1−2
1(1)−2(3)
,
1−2
1(8)−2(
2
9
)
⎠
⎟
⎟
⎟
⎟
⎟
⎞
=(5,1)
Therefore,
Equation of common tangent is-
S
1
−S
2
=0
(5x+y−6(
2
x+5
)−9(
2
y+1
)+13)−(5x+y−2(
2
x+5
)−16(
2
y+1
))=0
2
−6x−9y−13
+x+8y+13=0
4x−7y−13=0
Hence the point of contact is (5,1) and the equation of common tangent is 4x−7y−13=0.
Well, since it's already set up, all we do is divide -6 by 3/4 to get our x
So that means x=-8
Answer:
16y + 32
Step-by-step explanation:
Expand each term.
(y+6)² - (y-2)²
= (y+6)(y+6) - (y-2)(y-2)
= y² + 12y + 36 - (y² - 4y + 4)
Subtract the second group by changing each term's signs
= y² + 12y + 36 - y² + 4y - 4
Collect like terms
= 16y + 32
Domain: x² - 4 ≠ 0
+ 4 + 4
x² ≠ 4
√x² ≠ √4
x ≠ ±2
x ≠ -2 and x ≠ 2
(-∞, -2) ∨ (-2, 2) ∨ (2, ∞)
Range: y ≠ 1
(-∞, 1) ∨ (1, ∞)
Intervals: Increasing: (0.25 , ∞)
Decreasing: (-∞, 0.25)
Symmetry: X-axis: Not Symmetric
Y-axis: Not Symmetric
Origin: Not Symmteric
Extrema: Maximum Relative: x = 0
Minimum Relative: Nothing
