geometry

posted by .

What are the equations of the lines through (-5, -3) and passing at distance 2sqrt5 from (5,7)?

This problem is confusing? How do I solve this? Please I need your Help

  • geometry -

    all points 2√5 from (5,7) form a circle:

    (x-5)^2 + (y-7)^2 = 20

    Now you want a tangent to the circle that passes through (-5,-3). Must be a tangent, because any other line through the circle will come closer than 2√5 to the center.

    So, since lines through (-5,-3) with slope m are

    (y+3) = m(x+5), we need

    (x-5)^2 + (m(x+5)-3-7)^2 = 20
    (x-5)^2 + (mx+(5m-10))^2 = 20
    (m^2+1)x^2 + 10m(m-3)x + (5m-10)^2 - 20 = 0

    Now for the line to be tangent, the above equation must have a single root. That is, the discriminant must be zero:

    100m^2(m-3)^2 - 4(m^2+1)((5m-10)^2-20) = 0
    m = 1/2, 2

    so, the two lines are

    y = 1/2 (x+5) - 3
    y = 2(x+5) - 3

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Geometry :(

    this maths problem is wrecking my head.... 1.find the points of interesction of the line L and the circle k in the following L:x-2y=0 K:X Squared + y squared=25 I would solve both equations for y, and then you'll have a system with …
  2. Calculus

    Determine the equations of the following lines: (a) having slope ¡1 and passing through (0; 2) (b) parallel to y=2x-1 and passing through (¡2; 0) (c) passing through (1; 0) and (¡1; 3) (d) with x-intercept 6 and y-intercept 3.
  3. Precalculus check answers help!

    16. Find the distance between P (–2, 5) and the line with equation x – 3y + 4 = 0. (1 point) 17(sqrt10)/ 10 0 -17(sqrt10)/ 10 13(sqrt10)/ 10 ~ 17. Find the distance between the lines with equations 5x + 12y = 12 and y = –x + …
  4. Precalculus check answers help!

    16. Find the distance between P (–2, 5) and the line with equation x – 3y + 4 = 0. (1 point) 17(sqrt10)/ 10 0 -17(sqrt10)/ 10 13(sqrt10)/ 10 ~ 17. Find the distance between the lines with equations 5x + 12y = 12 and y = –x + …
  5. Math,Geometry

    What are the equations of the lines through (-5,-3) and passing at distance 2sqrt5 from (5,7) Please I really need your help. I do not know how to do this problem.
  6. Math,Geometry

    What are the equations of the lines through (-5,-3) and passing at distance 2sqrt5 from (5,7) Please I really need your help. I do not know how to do this problem.
  7. Analytic Geometry

    Find the equation of a circle passing through (3,7) and tangent to the lines x-3y+8=0 and y=3x. Please Help me solve this. Explain it to me slowly cause I am not really good with this
  8. Analytic Geometry

    Find the equation of a circle passing through (-1,6) and tangent to the lines x-2y+8=0 and 2x+y+6=0. Please I really need your help. I just do not know what to do.
  9. geometry

    write the equations of the horizontal and vertical lines through the given points. (5,6) Can someone please show me and explain to me how to solve this?
  10. Geometry

    Find the equations of lines parallel to and perpendicular to the line 3x - 2y = -1 and passing through the point (3, -2)

More Similar Questions