# Find equations for the tangent line and normal line to the circle at each given point.

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.
asked Jan 22, 2015 in CALCULUS

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 .

edited Jan 22, 2015 by Lucy

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 .

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

.