A circle has equation x^2+y^2+2x+4y-15=0 and the coordinates of T is (6,2). TA and TB are the tangents to the circle at the points A and B

Question

i) find radius and centre of circle

ii) find the coordinates A and of B

Answer 1

x^2+y^2+2x+4y-15=0

First, we'll get this into the standard form: (x + a)² + (y + b)² = r²

Collect the x and y terms on the left, and the constant on the right:

x² + 2x + y² + 4y = 15

Next, recognize, that adding some constants will make 2 perfect squares:

x² + 2x + 1 + y² + 4y + 4 = 15 + 5 (always do the same thing on both sides of an equation

rewrite in the standard format:

(x + 1)² + (y + 2)² = 20 (r is √20 = 2√5, and the center is at (-1, -2)

To find a tangent, draw a triangle where one side is the circle radius, the second side is the distance from the center to the given point (T), and the third side is the tangent line (TA or TB). The radius is perpendicular to a tangent line,and so we can use the Pythagorean Theorem to find the length of each tangent line.

We know the radius, and we can get the hypotenuse by finding the distance from the circle center to the point T.

Draw a picture (on graph paper) and this should make sense.