Find the length of perpendicular from ( 3, 2) on the line 4x- 6y -5= 0
Answers
Answer:
Let the foot of perpendicular from origin to a chord of circle x
2
+y
2
−4x−6y−3=0 has coordinates (h,k). Hence, slope of the line joining origin and foot of perpendicular is
h
k
. So, slope of the chord is −
k
h
.
Equation of the chord is y−k=−
k
h
(x−h)
hx+ky=h
2
+k
2
.....(1)
Now, the combined equation of the two lines joining the origin and two ends of the chord is obtained by homogenizing the equation of circle with the help of (1).
x
2
+y
2
+(−4x−6y)(
h
2
+k
2
hx+ky
)−3(
h
2
+k
2
hx+ky
)
2
=0
Since the two lines are perpendicular (given that they subtend right angle at origin), in the combined equation, coefficient of x
2
+ coefficient of y
2
=0
⇒[1−
h
2
+k
2
4h
−
(h
2
+k
2
)
2
3h
2
]+[1−
h
2
+k
2
6k
−
(h
2
+k
2
)
2
3k
2
]=0
⇒2(h
2
+k
2
)−(4h+6k)−3=0, which is the desired locus.
⇒2(x
2
+y
2
)−(4x+6y)−3=0, which is the required locus.
Hence, option 'A' is correct.
Step-by-step explanation:
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