If x2+y2+2x−4y+5=0, then the value of x+y is a) –1 b) 1 c) 2 d) ½
Answers
Given that the line ax+by=0 touches the circle x
2
+y
2
+2x+4y=0, i.e., the line is tangent to the circle.
Equation of circle can be written as-
(x+1)
2
+(y+2)
2
−1−4=0
⇒(x+1)
2
+(y+2)
2
=(
5
)
2
Therefore,
Centre of circle =(−1,−2)
Radius of circle =
5
As we know that perpendicular distance from a point (x
1
,y
1
) to the line ax+by+c=0 is given by-
d=
a
2
+b
2
∣ax
1
+by
1
+c∣
∴⊥ distance from the centre of circle to the line ax+by=0-
d=
a
2
+b
2
∣−a−2b∣
As we know that the perpendicular distance from the centre of circle to the line touching the circle is equal to the radius of circle.
∴
a
2
+b
2
∣−a−2b∣
=
5
Squaring both sides, we have
a
2
+b
2
a
2
+4b
2
+4ab
=5
⇒a
2
+4b
2
+4ab=5a
2
+5b
2
⇒4a
2
+b
2
−4ab=0
⇒(2a−b)
2
=0
⇒2a−b=0.....(1)
Answer:
(b) 1
Step-by-step explanation:
x2 + y2 + 2x - 4y + 5 = 0
or, (x + 1)^2 + (y - 2)^2 = 0
or, (x + 1)^2 = - (y - 2)^2
or, x + 1 = - y + 2
or, x + y = 1
So, Ans:- (b) 1
Hope it helps!