Question

Point by point, a dilation transforms the circle x²−6x+y²−8y = −24 onto the circle 9x²−14x+y²−4y = −44. The circles have two common external tangent lines, which meet at the dilation center. Find the size of the angle formed by these lines, and write an equation for each line.

Answer 1

The dilation factor can be found using the ratio of the radii of the two circles. The common tangent lines to these circles intersect at the dilation center and have opposite slopes. Equations for these lines can be written using the point-slope form.

Step-by-step explanation:

To solve this problem, we first need to find the centers of both circles and the dilation factor. The general equation of a circle is (x-a)²+(y-b)²=r². Thus, from the given equations, we can find the centers (h1, k1) and (h2, k2), by comparing coefficients.

For the first circle center (h1, k1) = (3, 4) and for the second circle, center (h2, k2) = (7/6, 2). The dilation factor, d, can be found using the ratio of the radii of the two circles, which can be obtained by completing the squares in the given equations.

The common external tangent lines to these circles that intersect at the dilation center can be found using the formula for the angle between two lines based on their slopes. Assuming the circles are symmetrical about the x-axis, the tangent lines will be symmetric too, so their slopes will have the same absolute values but opposite signs.

We can now write equations for these lines using the point-slope form of lines, using the common dilation center as the point and the tangents we found as the slopes.

Learn more about Dilation of Circles