question : x+y +2=0, x^2+ y^2-10y=0 find the angle between the curves
Answers
Step-by-step explanation:
SWER
Solving xy=2 and x
2
+4y=0
y=
4
−x
2
But xy=2
⇒x(
4
−x
2
)=2
⇒x
3
=−8
⇒x=−2
⇒y(
4
−x
2
)=
4
−2
2
=−1
point of intersection is p(−2,−1)
xy=2 or y=
x
2
dx
dy
=
x
2
−2
m
1
=(
dx
dy
)
p
=
4
−2
=
2
−1
⇒x
2
+4y=0
⇒y=
4
−x
2
⇒
dx
dy
=
4
−2x
=
2
−x
m
2
=(
dx
dy
)
p
=
−2
−2
=1
Let ϕ be the angle between the given curves
tanϕ
∣
∣
∣
∣
∣
1+m
1
m
2
m
1
−m
2
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
∣
∣
∣
∣
1+(−
2
1
)1
2
−1
−1
∣
∣
∣
∣
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
∣
∣
∣
2
1
2
−3
∣
∣
∣
∣
∣
∣
∣
∣
=3
Answer:
Step-by-step explanation:
We are given equations of two circles x2+y2−10y=0and x2+y2+6y=0
We know that the general equation of a circle is x2+y2+2gx+2fy+c=0
Its centre and radius is given by (−g,−f) and g2+f2−c−−−−−−−−−√
Let's consider the circle x2+y2−10y=0
Here 2g=0 and 2f=−10
Hence g=0 and f=−5
Therefore its centre is (−g,−f)=(0,5)
And radius is 02+(−5)2−0−−−−−−−−−−−−√=25−−√=5units
Let's consider the circle x2+y2+6y=0
Here 2g=0 and 2f=6
Hence g=0 and f=3
Therefore its centre is (−g,−f)=(0,−3)
And radius is 02+(3)2−0−−−−−−−−−−√=9–√=3units
Let's find the distance between the centres
⇒(x1−x2)2+(y1−y2)2−−−−−−−−−−−−−−−−−−√⇒(0−0)2+(5−(−3))2−−−−−−−−−−−−−−−−−√⇒(8)2−−−√=8units
Sum of radii = 5 + 3 =8 units
Therefore the distance between the centres is equal to the sum of their radii
Hence the circles touch each other externally
seo images
Now the tangents touch the circle x2+y2−10y=0at C and D and touch the circle x2+y2+6y=0 at A and B
Let PQR be the triangle formed
By the property ,
The length of tangents from an external point to a circle are equal.
⇒PA=PB……..(1)
⇒PC=PD …….(2)
From this
We get that AC=BD ……..(3)
And AQ=BR ……..(4)
Adding (1) and (4) we get
⇒PA+AQ=PB+BR⇒PQ=PR
Therefore we get that the two sides of the triangle PQR are equal .
Hence it is an isosceles triangle
The correct option is a.
Note: The tangent always touches the circle at a single point.
It is perpendicular to the radius of the circle at the point of tangency
It never intersects the circle at two points.