write an equation of the line that is tangent to the circle at the given point.x^2+y^2=24

Question

x^2+y^2=24; (-2*square root of 2, -4)

Answer 1

A line tangent to a circle and radius to the point of tangency are perpendicular.

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Answer 2

The circle is x² + y² = 24 and the point is (- 2√2, - 4).

Differentiate with respect to x.

2x + 2y y ' = 0

x + y y ' = 0

y y ' = - x

⇒ y ' = - x/y.

When, x = - 2√2 and y = - 4,  y ' = - ( - 2√2 )/- 4 = - 1/√2.

y ' = - 1/√2.

This is the slope (m ) of the tangent line to the circle at (- 2√2, - 4).

Slope - intercept form line equation is y = mx + b, where m is slope and b is y - intercept.

Slope(m) = - 1/√2.

Now the tangent line equation is y = (- 1/√2)x  + b.

Find the y - intercept by substituting the the point in the tangent line equation say (x, y) = (- 2√2, - 4).

- 4 = (- 1/√2)(- 2√2) + b

b = - 4 - 2

⇒ b = - 6.

The tangent line equation  is y = (- 1/√2)x - 6.