The straight lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to

Question

The straight lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle, then the radius of this circle is

(A) 1/4 
(B) 3/4 
(C) 4/3 
(D) 5/3 
(E) 2

sumitkrocks · Answer

The straight lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle, then the radius of this circle is

As the slop of the two distinct lines are same; hence, we can conclude that the distance between these two line will be equal to diameter

Calculating Distance = (4+ \frac{7}{2} )/ \sqrt{3^2+ 4^2}  =  \frac{3}{2}

Radius=  \frac{3}{4}

Option B is correct IMO

Jihyo · Answer

When two equations are in this pattern- ax+by+c1=0 and ax+by+c2=0 -they will be parallel lines.
The formula of getting the distance between two parallel line is:
|C1-C2|/\sqrt{a^2+b^2}

We have two equations in this question: 3x-4y+4=0.....(1)
 6x-8y-7=0
 =>3x-4y-7/2=0.....(2)
Here,C1=4, C2=-7/2, a=3,b=-4 
so, |4+7/2|/ \sqrt{9+16}=3/2

Now if the distance is 3/2, the radius will be 3/2*1/2=3/4
(The distance between two parallel tangents of a circle is the diameter of that circle)

ANSWER: B

Nidzo · Answer

To solve we need half the distance between the tangents.

3x - 4y + 4 = 0 becomes  y = 3/4x + 1   (1)

and 6x - 8y - 7 = 0 becomes  y = 3/4x - 7/8   (2)

To find the distance between the two points create a perpendicular line to line  (1)  with the y-intercept 1. 
With this one can find the intersecting point between the perpendicular line and line  (2) , thus giving you two parallel points with which to measure the distance between the two lines.

Perpendicular line of line  (1)  with y-intercept 1 will have the formula:  y = -4/3x + 1   (3)

Make lines  (2)  and  (3)  equal to one another to find their intersecting x value:
 
 3/4x - 7/8 = -4/3x + 1 
15/8 = 25/12x
180 = 200x
x = 9/10

Plug x into  (2)  and solve for y

y = 3/4(9/10) - 7/8
y = 27/40 - 35/40
y = -8/40
y = -1/5

The two parallel points are  (0 ; 1)  and  (9/10 ; -1/5)

Plug them into the distance formula  \sqrt{(x1-x2)^2 + (y1-y2)^2}

\sqrt{(0 - 9/10)^2 + (1 + 2/10)^2}

\sqrt{(-9/10)^2 + (12/10)^2}

\sqrt{ 81/100 + 144/100 }

\sqrt{ 225/100 }

15/10

3/2  is the full length of the distance between the two lines, half is the radius of the circle therefore radius is  3/4

Answer B

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The straight lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to

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The straight lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to

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