Question

Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, tangent line, and normal line.

Two circles of radius 4 are tangent to the graph of y^2 = 4x at the point (1, 2). Find equations of these two circles.

Answer 1

Step 1:

The equation is and the point is .

Consider .

Differentiate on each side with respect to .

Slope of the tangent line at .

Slope of the tangent line is .

Step 2:

Point slope form of line equation is .

Substitute and in the above equation.

Tangent line is .

Normal line is perpendicular to tangent line then

Slope of tangent line*slope of normal line is equal to .

Point slope form of line equation is .

Substitute and in the above equation.

Normal line equation is .

Answer 2

Contd..

Step 3:

Equation of circle with center and radius is .

Differentiate on each side with respect to .

Substitute .

Substitute in the above equation.

Step 4:

Substitute in the circle equation

Circle passes through the point .

Roots of the quadratic equation is .

Then,

Therefore and .

Answer 3

Contd..

Step 5:

Substitute in .

Substitute in .

Circle equations are and

.

Step 6:

Graph.

Graph both the circle equations, curve, tangent line, normal line and point.

Solutions:

Tangent line equation is .

Normal line equation is .

Circle equations are and

.