Question

the center of a circle is at the origin on a coordinate grid. a line with a positive slope intersects the circle at (0,7). which statement must be true?
the circle has a radius greater than 7.
the circle has a radius equal to 7.
the slope of the line is equal to 7.
the slope of the line is not equal to 7.

Explanation:

Step1: Recall the distance formula for the radius

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For a circle centered at the origin \((0,0)\) and a point \((0,7)\) on the circle, the radius \(r=\sqrt{(0 - 0)^2+(7 - 0)^2}=7\). So the radius of the circle is 7, which rules out the option “The circle has a radius greater than 7”.

Step2: Analyze the slope of the line

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). A line with a positive slope passing through \((0,7)\) and another point \((x,y)\) on the circle (\(x
eq0\)) has \(m=\frac{y - 7}{x-0}\). Since \(x
eq0\) (if \(x = 0\), the line is vertical and slope is undefined) and \(y
eq14\) (from the circle equation \(x^{2}+y^{2}=49\)), \(m
eq7\).

Answer:

The slope of the line is not equal to 7.