Question

5.

Answer 1

Answer:  (x - 4)² + (y - 7)² = 9

Explanation:

The equation of a circle is: (x - h)² + (y - k)² = r²    where

• (h, k) is the center
• r is the radius

Given: (h, k) = (4, 7)

Find the intersection of the given equation and the perpendicular passing through (4, 7).

3x - 4y = -1

     -4y = -3x - 1

        $y=\dfrac{3}{4}x-1$

              $m=\dfrac{3}{4}$      -->      $m_{\perp}=-\dfrac{4}{3}$

$y-y_1=m_{\perp}(x-x_1)\\\\y-7=-\dfrac{4}{3}(x-4)\\\\\\y=-\dfrac{4}{3}x+\dfrac{16}{3}+7\\\\\\y=-\dfrac{4}{3}x+\dfrac{37}{3}$

Use substitution to find the point of intersection:

$x=\dfrac{29}{5}=5.8,\qquad y=\dfrac{23}{5}=4.6$

Use the distance formula to find the distance from (4, 7) to (5.8, 4.6) = radius

$r=\sqrt{(5.8-4)^2+(4.6-7)^2}\\\\r=\sqrt{3.24+5.76}\\\\r=\sqrt9\\\\r=3$

Input h = 4, k = 7, and r = 3 into the circle equation:

(x - 4)² + (y - 7)² = 3²

(x - 4)² + (y - 7)² = 9