Geometry

posted by .

What is the minimum distance between any point on the circle x^2 + y^2 = 25 and the line y = -\frac{3}{4}x + \frac{75}{4} ?

  • Geometry -

    The distance from (h,k) to the line ax+by+c=0 is |ah+bk+c|/√(a^2+b^2)

    y = 3/4 x + 75/4 is
    3x - 4y + 75 = 0

    we know that if (h,k) is on the circle,

    k=√(25-h^2)

    so, the distance s from (h,k) to 3x-4y+75=0 is

    s = (3h-4√(25-h^2)+75)/5
    find where s'=0 and that will give you h, and you can figure k.

  • Geometry -

    Or, you know that at (h,k) on the circle, the slope is -h/k.

    The slope of the line is 3/4
    The minmum distance will be a line perpendicular, over to the circle.

    So, we want a line with slope -4/3 = -h/k, and we want it in QII, since that's where the line is.

    Looks like (-4,3) is the point we want.



    Just for grins, run through the other solution and see whether it agrees.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Geometry

    What is the minimum distance between any point on the circle x^2 + y^2 = 25 and the line y = -\frac{3}{4}x + \frac{75}{4} ?
  2. Algebra

    How many ordered pairs of solutions (a, b) are there to \frac{a}{b} - \frac{b}{a} - \frac{2}{a} - \frac{2}{b} = 0, where a and b are integers from -100 \leq a,b \leq 100?
  3. Quantum Physics

    Let |\psi\rangle=\frac{1-i}{2}|0\rangle-\frac{1+i}{2}|1\rangle and |\phi\rangle=\frac{2+i}{3}|0\rangle-\frac{2i}{3}|1\rangle. What is \langle \psi\,|\,\phi\rangle?
  4. Algebra

    a and b are positive numbers that satisfy the equation \frac {1}{a} - \frac {1}{b} = \frac {1}{a+b} . Determine the value of \frac {a^6}{b^6} + \frac {b^6} {a^6} .
  5. Algebra

    If a, b and c are non-zero reals such that a + b + c = 11 and \frac {1}{a} + \frac {1}{b} +\frac {1}{c} = 0, what is the value of a^2 + b^2 + c^2?
  6. Algebra

    If a, b and c are non-zero reals such that a + b + c = 11 and \frac {1}{a} + \frac {1}{b} +\frac {1}{c} = 0, what is the value of a^2 + b^2 + c^2?
  7. Trigonometry

    Given \tan \theta = -\frac{4}{3}, where \frac{\pi}{2} < \theta < \pi, what is the value of \frac{1}{\sin \theta + \cos \theta}?
  8. Physics

    The equation describing the (r, \theta ) coordinates of points along a single field line of a magnetic dipole is r=R_0 \sin ^2(\theta ) where \theta =0 is in the direction of the dipole moment and R_0 is a constant which is different …
  9. Calculus

    A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks d miles apart, the concentration of the combined deposits on the line joining them, …
  10. math

    Find an integer $x$ such that $\frac{2}{3} < \frac{x}{5} < \frac{6}{7}$

More Similar Questions