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# What are the lengths of the tangent lines when they touch the circle x squared plus y squared -10x plus 8y plus 5 equals 0 from the point 5 4 on the Cartesian plane?

Updated: 12/20/2022

Wiki User

7y ago

A circle with centre (x0, y0) and radius r has the formula:

(x - x0)² + (y - y0)² = r²

Completing the squares:

x² + y² - 10x + 8y + 5 = 0

→ x² -10x + 25 - 25 + y² + 8y + 16 - 16 + 5 = 0

→ (x - 5)² - 25 + (y + 4)² - 16 + 5 = 0

(x - 5)² + (y + 4)² = 36 = 6²

→ The circle has centre (5, -4) and radius 6.

A tangent to the circle forms a right angle with the radius of the circle that meets the tangent.

Joining the centre of the circle to the point (5, 4) will form the hypotenuse of the triangle with the radius and the tangent as the other two sides.

The length of the hypotenuse can be calculated using Pythagoras:

hypotenuse² = (5 - 5)² + (-4 - 4)² = 0 + 8² = 8²

Thus the length of the tangents from the point (5, 4) can be calculated using Pythagoras:

→ 6² + tangent² = 8²

→ tangent² = 8² - 6² = 64 - 36 = 28

→ tangent = √28 = √(4×7) = √4 √7 = 2 √7

Each tangent is 2 √7 units long.

Wiki User

7y ago

Wiki User

7y ago

They are 2*sqrt(7) units long.