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the circle is tangent to the line y=5x 3 at the point (2,7) and the circle is on the line y=-1/2x 19. FIND THE EQUATION OF THE CIRCLE

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The equation of the circle in standard form is; (x - 7)² + (x + 6)² = 26

The steps that can be used to find the equation of the circle can be presented as follows;

The radius from the center of the circle to the tangent is perpendicular to the tangent, therefore;

The equation of the radius of the circle can be found as follows;

Slope of the radius of the circle = 1/5

Equation of the radius is; (y - (-7)) = (1/5) × (x - 2)

y + 7 = (1/5) × (x - 2)

5·y + 35 = x - 2

x = 5·y + 35 + 2

x = 5·y + 37

The coordinates of the center is therefore, the location of the intersection of the radius and the line x = 2·y + 19, which can be found as follows;

Equating the equation of the radius to the equation of the line passing through the center, we get;

5·y + 37 = 2·y + 19

5·y - 2·y = 19 - 37

3·y = -18

y = -18/3

y = -6

x = 5·y + 37

x = 5 × (-6) + 37

x = 7

The coordinates of the center of the circle is; (7, -6)

The length of the radius of the circle is therefore;

r = √((2 - 7)² + (-7 - (-6))²)

r = √(26)

The equation of the circle is therefore;

(x - 7)² + (y - (-6))² = 26

(x - 7)² + (y + 6)² = 26

The complete question, obtained from a similar question found through can be presented as follows;

The circle is tangent to the line y = 3 - 5·x at the point (2, -7) whose center is on the line x = 2·y + 19 find the equation of the circle

User Binyamin
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