A(-3,-2) B(8,4)are the ends of the diameter of a circle.
the tangent of the circle at point B is -11x+6y+64=0
the line L passes through A and the origin. line L has a gradient of 2/3
what is the coordinates that the line L and tangent at point B cross??
(this question had 4 parts, ive done the first 3 and i think ive got the calculations right. im just stuck on the 4th part)
please help thanks
To find the coordinates where line L and the tangent at point B cross, you need to find the intersection point of these two lines. Here's how you can do it step by step:
1. Find the equation of line L passing through point A(-3,-2) and the origin (0,0) with a gradient of 2/3.
- The equation of a line in point-gradient form is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the gradient.
- Substituting the values, the equation of line L is: y - (-2) = (2/3)(x - (-3))
Simplifying, we get: y + 2 = (2/3)(x + 3)
2. Rewrite the equation of the tangent at point B in the standard form (Ax + By + C = 0) form.
- The given equation of the tangent is: -11x + 6y + 64 = 0
- We want to rearrange it to the standard form, so let's isolate y and rewrite: 6y = 11x - 64 -> y = (11/6)x - (64/6) -> y = (11/6)x - 32/3
- Multiply through by 6 to get rid of the fraction: 6y = 11x - 32
3. Now, you have two equations:
- Equation of line L: y + 2 = (2/3)(x + 3)
- Equation of the tangent: 6y = 11x - 32
4. Set the two equations equal to each other to find the intersection point.
- Substitute y from the first equation into the second equation:
(2/3)(x + 3) + 2 = (11/6)x - 32
- Simplify and solve for x:
(2/3)x + 2 + 2 = (11/6)x - 32
(2/3)x + 4 = (11/6)x - 32
Multiply through by 6 to eliminate fractions:
4x + 24 = 11x - 192
Rearrange and solve for x:
11x - 4x = 24 + 192
7x = 216
x = 216/7
5. Substitute the value of x back into either of the original equations to find the value of y:
- Using the first equation, substitute x = 216/7:
y + 2 = (2/3)(216/7 + 3)
y + 2 = (2/3)(216/7 + 21/7)
y + 2 = (2/3)(237/7)
y + 2 = (2/3)(237)/7
y + 2 = (474/3)/7
y + 2 = 474/21
y = 474/21 - 2
y = 474/21 - 42/21
y = 432/21
So, the coordinates where line L and the tangent at point B cross are approximately (216/7, 432/21).