Question

if the length of the particular drawn from the point (1,5) on the line 3x-4y+k=0 is 2√13 find the value of k

Answer 1

Answer:

Since the given line touches the given circle,

the length of the perpendicular from the center (2,4) of the circle to the line

3x−4y−k=0 is equal to the radius

4+16+5

=5 of the circle.

⇒

9+16

3×2−4×4−k

=±S⇒k=15 [∵k>0]

Now, equation of the tangent at (a,b) to the given circle is

xa+yb−2(x+a)−4(y+b)−5=0

⇒(a−2)x+(b−4)y−(2a+4b+5)=0

If it represents the given lines 3x−4y−k=0

then,

3

a−2

=

−4

b−4

=

k

2a+4b+5

=l (say)

Then, a=3l+2,b=4−4l and 2a+4b+5=kl ...(1)

⇒2(3l+2)+4(4−4l)+5=15l (∵k=15)

⇒l=1⇒a=5,b=0 Then, k+a+b=20

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