The line \(y = \frac{3x + 15}{4}\)intersects the circle \(x^2 + y^2 = 36\) at A and B. Find the length of chord AB.
Please help soon. Thanks! :)
3x+15 = y sub that in to the circle equation
x^2 + [(3x+15)/4]^2 = 36
x^2 + [(9x^2 +90x+225)/16 ] = 36
16x^2 + 9x^2 + 90x + 225 = 576
25x^2 + 90x - 351 = 0 Use quadratic formula to find x = 2.35692 and −5.95692 (these are 9/5 +- 12 sqrt5/5 )
Now you can use these values of x to find the y values by substituting into the line equation
THEN you can use the distance formula to calculate the length sqrt { (x1-x2)^2 + (y1-y2)^2 }
Is there an easier way??? Probably........
Thanks, EP
Here's another method....albeit....maybe not any easier .....LOL!!!!!
The line intersects the y axis at 15/4 = 3.75
And it intersects the x axis at x = -5
The distance between these two points is sqrt ( 5^2 + 3.75^2 ) = 6.25
Call the distance between the y intercept and the point where the line intersects the upper-most part of the circle, a
Call the distance from the x intercept of the line and the point where the line intersects the "lower" part of the circle , b
The distance between the y intercept and the top of the circle = 6 -3.75 = 2.25
And the distance from the y intercept to the bottom part of the circle = 3.75 + 6 = 9.75
Likewise......the distance from the x intercept to the leftmost point of the circle =1
And the distance from the x intercept to the rightmost part of the circle =11
By the intercepting chord theorem we have this system
(a) (6.25 + b) = (2.25) (9.75)
(a+ 6.25) (b) = (1) (11) simplify
6.25a + ab = 21.9375 (1)
6.25b + ab = 11 (2) subtract these
6.25 ( a - b) = 10.9375
(a - b) = 10.9375/6.25
a - b = 1.75
a = b + 1.75
Subbing this into (2) we get that
(b + 1.75 + 6.25) ( b) =11
( b + 8) b =11
b^2 + 8b - 11 = 0
b^2 + 8b = 11
b^2 + 8b + 16 = 11 + 16
(b + 4)^2 = 27 take the positive root
b + 4 = √27 = 3√3
b = 3√3 - 4
And
a = b + 1.75 = 3√3 - 2.25
So.....the length of chord AB =
b + 6.25 + a =
3√3 - 4 + 6.25 + 3√3 - 2.25 =
6√3 + 6.25 - 6.25 =
6√3
Thank you Electric Pavlov and CPhill!!! Both your ways were good, I was kinda lost on your way CPhill. I'm not too good at piecing together something that large. I was trying something along the lines of Electric Pavlov's solution, but coudn't figure out how to finish it. Thanks both of you. Stay safe and healthy!!!