Write an equation for each circle described below. a circle with its center in quadrant I, radiu

Write an equation for each circle described below. a circle...

Key Concepts

Tangency Condition

When a circle is tangent to a line, it touches the line at exactly one point. The distance from the center of the circle to a tangent line is equal to the radius of the circle. This relationship is fundamental when determining unknown parameters of the circle based on given tangency conditions.

Coordinate Plane and Quadrants

The location of a circle’s center in a specific quadrant of the Cartesian coordinate plane influences the signs of its coordinates. Being in quadrant I means both the x-coordinate and y-coordinate of the center are positive, which is crucial when determining the correct values for the center in problems involving tangency to lines.

Standard Equation of a Circle

The standard form for the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the circle’s center and r is its radius. This form encapsulates all the geometric information about the circle, enabling analyses of its properties and relationships with other figures.

Radius of a Circle

The radius is the distance from the center of the circle to any point on the circle. It is a key parameter that defines the size of the circle and appears squared in the circle’s standard equation, signifying the fixed distance that all points on the circle share from the center.

Transcript

00:01 In this problem, we're asked to find the equation of a circle that is in quadrant 1, radius of 5 units, and tangent to the lines x equals 2 and y equals 3.

00:11 So, first of all, to understand what quadrant we are in, we are in this top right quadrant, which means both our x and y values for our center will be positive.

00:24 And we're given a radius of five units and these tangent lines x equals two and y equals three and what that means is for our circle it touches these two lines at its edges so it's the farthest that our circle goes from the center so to find the equation of a circle we have the quantity x minus h squared plus the quantity y minus k squared equals r squared and our center is given by h comma k and so in our equation we have our value for r squared and we have x and y we just leave those as variables so all we need to do is find the center of our circle so if we come down here to a slightly larger graph um here we have, again, our tangent lines, x equals two and y equals three.

01:34 And we can use these tangent lines to find our center.

01:38 So we know that where our circle touches this tangent line is the farthest that our circle goes, which means wherever our circle touches the tangent line, that has to be five units from the center.

01:59 So what we can do is starting where our two tangent lines meet up, we can count five units out from both of these lines to find our center.

02:13 So if we go ahead and do that and we start from y equals three, we can find how.

02:20 High our radius goes from this tangent line.

02:28 So if we go up one, two, three, four, five, this is the height of our center...

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