find shortest distance between
y^(2)=4x and (x-6)^(2)+y^(2)=1
Parabola
Answers
Step-by-step explanation:
Correct option is
B
5
Since the shortest distance between the two curves happens to be at the normal which is common to both the cuves.
Therefore
The normal to the curve y
2
=4x at (m
2
,2m) is given by:
(y−2m)=−m(x−m
2
)
i.e., y−2m=−mx+m
3
i.e., y−2m+mx−m
3
=0
i.e., y+mx−2m−m
3
=0
And the normal to the curve y
2
=2x−6 at (
2
1
m
2
+3,m) is given by:
(y−m)=−m(x−
2
1
m
2
−3)
i.e., y−m=−mx+
2
1
m
3
+3m
i.e., y−m+mx−
2
1
m
3
−3m=0
i.e., y+mx−4m−
2
1
m
3
=0
These two normals are common if:
y+mx−2m−m
3
= y+mx−4m−
2
1
m
3
i.e., −2m−m
3
= −4m−
2
1
m
3
i.e., m
3
−4m=0
i.e., m(m
2
−4)=
i.e., m(m+2)(m−2)=
Therefore, m=0,2,−2
Thus, the points are: (4,4) and (5,2)
And the distance is: d=
(5−4)
2
+(2−4)
2
i.e., d=
1+4