Question

Prove algebraically that the straight line with equation x =2y+5 is a tangent to the circle with equation x ² +y ²

Answer 1

The intersection point is only one. Then the equation of the line is tangent to the circle at point (1, -2).

What is a circle?

It is a locus of a point drawn an equidistant from the center. The distance from the center to the circumference is called the radius of the circle.

Prove algebraically that the straight line with equation x = 2y + 5 is a tangent to the circle with equation x² + y² = 5.

x = 2y + 5 ...1

x² + y² = 5 ...2

If the intersection of the point of the circle and line is one. Then the line is tangent to the circle.

Then from equations 1 and 2, we have

               (2y + 5)² + y² = 5

4y² + 25 + 20y + y² - 5 = 0

            5y² + 20y + 20 = 0

    5y² + 10y + 10y + 20 = 0

     5y (y + 2) + 10(y + 2) = 0

              (5y + 10)(y + 2) = 0

                                    y = -2, -2

Then the value of y is unique then the value of x will be unique.

The value of x will be

x = 2(-2) + 5

x = -4 + 5

x = 1

The intersection point is only one. Then the equation of the line is tangent to the circle at point (1, -2).

More about the circle link is given below.

brainly.com/question/11833983

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