Math, asked by sanskritisamdhian, 8 months ago

question : x+y +2=0, x^2+ y^2-10y=0 find the angle between the curves

Answers

Answered by sudiptamaji2007
4

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

Answered by pgirivardhan
1

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.

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