Question

Is the tangent at (1,7) to the curre x²=y-6 touches
the circle x2+y2+16x+12y+c=0 , then the value of. C is
(a) 95
(6) l95 (C) 185 (d) 85​

Answer 1

Answer:

Step-by-step explanation:

Given equation of curve is x  

2

=y−6

⇒y=x  

2

+6

⇒  

dx

dy

​  

=2x

Slope of tangent is m=  

∣

∣

∣

∣

∣

​  

 

dx

dy

​  

 

∣

∣

∣

∣

∣

​  

 

(1,7)

​  

=2×1=2

Equation of tangent at the point (1,7) is given by

(y−7)=2(x−1)

⇒y−7=2x−2

⇒2x−y+5=0

Given equation of circle is  

x  

2

+y  

2

+16x+12y+c=0

⇒(x+8)  

2

+(y+6)  

2

+c−64−36=0

⇒(x+8)  

2

+(y+6)  

2

=100−c

If the tangent touches the circle, then

Distance from centre = radius

⇒ Distance of (−8,−6) from 2x−y+5=0 is the radius

d=  

∣

∣

∣

∣

∣

​  

 

4+1

​  

 

2(−8)−(−6)

​  

 

∣

∣

∣

∣

∣

​  

=  

∣

∣

∣

​  

 

5

​  

 

∣

∣

∣

​  

 

⇒Radius=  

100−c

​  

=  

5

​  

 

⇒c=95