Use the discriminant to find the gradients of the lines that pass through the point (7,1) and are tangent to the circle x^2 + y^2 = 25.

Question

Working out please!

Step by step..

Thank you!

Answer 1

I am not sure what is meant by using the discriminant to find the gradient; however, here is my solution to the problem.

A point on the circle is (x,sqrt(25-x2)

At that point the slope of the tangent line to the circle is -x/sqrt(25-x²).

Then the equation of the tangent line is -x/sqrt(25-x²) = [sqrt(25-x²) - 1]/(x-7)

Cross multiply

-x² + 7x = 25 - x² - sqrt(25-x²) => 7x - 25 = sqrt(25-x²)

and then

49x² - 350x + 625 = 25 - x² =>50x² -350x + 600 = 0

which has solutions x=3 and x = 4.

You now need a figure to see that the points of tangency are (3.4) and (4,-3) from which you can compute the slope of each tangent!,