Question

find the equations tangents to the circle x^2 + y^2 + 2x - 2y - 3 = 0 which are perpendicular to 3x - y + 4=0​ solve this problem

Answer 1

Step-by-step explanation:

Equation of ellipse-

2x  

2

+y  

2

=8

⇒  

4

x  

2

 

​  

+  

8

y  

2

 

​  

=1

Here,

a  

2

=4

b  

2

=8

Given equation of tangent-

x−2y−4=0

y=  

2

x

​  

−  

2

4

​  

 

Now,

(i) Equation of tangent parallel to y=  

2

x

​  

−  

2

4

​  

 is-

y=  

2

x

​  

+  

2

k

​  

.....(1)

Here

m=  

2

1

​  

 

c=  

2

k

​  

 

Equation (1) is tangent to the ellipse if

c  

2

=a  

2

m  

2

+b  

2

 

(  

2

k

​  

)  

2

=4×  

4

1

​  

+8

4

k  

2

 

​  

=9

⇒k=±  

36

​  

=±6

Hence the equation of tangents are y=  

2

x

​  

±  

2

6

​  

⇒2y=x±6

(ii) Equation of tangent perpendicular to y=  

2

x

​  

−  

2

4

​  

 is-

y=−2x+  

2

k

​  

.....(2)

Equation (1) is tangent to the ellipse if

(  

2

k

​  

)c  

2

=a  

2

m  

2

+b  

2

 

4

k  

2

 

​  

=4×4+8

4

k  

2

 

​  

=96

k=±  

96

​  

=±4  

6

​  

 

Hence the equation of tangents are y=−2x±4  

6