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# What is the length of the tangent line from the point -2 3 to a point where it touches the circle of x2 plus y2 plus 6x plus 10y -2 equals 0?

Updated: 10/17/2022

Wiki User

6y ago

Best Answer

The tangent of a circle is perpendicular to the radius to the point of contact (Xc, Yc).

The point (-2, 3), the centre of the circle (Xo, Yo) and the point of contact of the tangent (Xc, Yc) form a right angle triangle.

The leg from the point (-2, 3) to the point of contact (Xc, Yc) is the required length

The leg from the centre of the circle (Xo, Yo) to the point of contact (Xc, Yc) has length equal to the radius (r) of the circle

The hypotenuse is the length between the point (-2, 3) and the centre of the circle (Xo, Yo).

To solve this:

1. Find the centre (Xo, Yo) of the circle, and its radius r.
2. Use Pythagoras to find the length between the point (-2, 3) and the centre of the circle (Xo, Yo)
3. Use Pythagoras to find the length between the point (-2, 3) and the point of contact (Xc, Yc) of the tangent - the required length.

Hint: a circle with centre (Xo, Yo) and radius r has an equation of the form:

(x - Xo)² + (y - Yo)² = r²

Have a go at solving it now you know how, before reading the solution below:

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Circle:

x² + y² + 6x + 10y - 2 = 0

→ x² + 6x + y² + 10y - 2 = 0

→ (x + (6/2))² - (6/2)² + (y + (10/2))² - (10/2)² - 2 = 0

→ (x + 3)² - 9 + (y + 5)² - 25 - 2 = 0

→ (x + 3)² + (y + 5)² = 36 = 6²

→ Circle has centre (-3, -5) and radius 6

Line from centre of circle (-3, -5) to the given point (-2, 3):

Using Pythagoras to find length of a line between two points (x1, y1) and (x2, y2):

length = √((x2 - x1)² + (y2 - y1)²)

To find length between given point (-2, 3) and centre of circle (-3, -5)

→ length = √((-5 - -2)² + (-3 - -3)²)

= √((-3)² + (-6)²)

= √45

Tangent line segment:

Using Pythagoras to find length of tangent between point (-2, 3) and its point of contact with the circle:

centre_to_point² = tangent² + radius²

→ tangent = √(centre_to_point² - radius²)

= √((√45)² + 6²)

= √(45 + 36)

= √81

= 9

The length is 9 units.

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