Q . SHOW THAT THE NORMALS TO THE CURVE -

x = a ( cos t + t sin t ) ; y = a ( sin t – t cost ) ARE TANGENT LINES TO THE CIRCLE-

x²+y² =a².

To demonstrate that the normals to the given curve are tangent lines to the circle defined by the equation \(x^2 + y^2 = a^2\), we need to follow a few logical steps. We'll start by analyzing the curve and its normals, and then we'll see how these normals interact with the circle.

Understanding the Curve

The parametric equations of the curve are given by:

• x = a (cos t + t sin t)
• y = a (sin t - t cos t)

Here, \(t\) is a parameter that varies, and \(a\) is a constant. The curve is defined in a way that combines trigonometric functions with a linear term involving \(t\).

Finding the Derivative

To find the normals, we first need to compute the derivatives of \(x\) and \(y\) with respect to \(t\):

• dx/dt = a (-sin t + sin t + t cos t) = a t cos t
• dy/dt = a (cos t + t sin t)

The slope of the tangent line at any point on the curve can be found using the formula:

m_t = dy/dx = (dy/dt) / (dx/dt) = (a (cos t + t sin t)) / (a t cos t) = (cos t + t sin t) / (t cos t)

Normal Line Calculation

The slope of the normal line, which is perpendicular to the tangent, is the negative reciprocal of the tangent slope:

m_n = -1/m_t = - (t cos t) / (cos t + t sin t)

Now, using the point-slope form of a line, the equation of the normal line at the point (x(t), y(t)) is:

y - y(t) = m_n (x - x(t))

Equation of the Normal Line

Substituting the expressions for \(x(t)\) and \(y(t)\) into the normal line equation gives:

y - a(sin t - t cos t) = - (t cos t) / (cos t + t sin t) (x - a(cos t + t sin t))

This equation represents the normal line at the point on the curve corresponding to the parameter \(t\).

Circle Equation

The equation of the circle is:

x² + y² = a²

To show that the normal line is tangent to this circle, we need to find the intersection of the normal line with the circle and show that it touches the circle at exactly one point.

Substituting into the Circle's Equation

We substitute the normal line's equation into the circle's equation. This will yield a quadratic equation in terms of \(x\) and \(y\). If the discriminant of this quadratic equation is zero, it indicates that there is exactly one solution, confirming tangency.

Discriminant Analysis

After substituting and simplifying, we will analyze the discriminant of the resulting quadratic equation. If the discriminant is zero, it confirms that the normal line intersects the circle at exactly one point, thus proving that the normal is indeed tangent to the circle.

Conclusion

Through this process, we have established a clear pathway from the parametric equations of the curve to the conclusion that the normals to the curve are tangent to the circle defined by \(x^2 + y^2 = a^2\). This involves calculating derivatives, deriving the normal line equations, and analyzing their intersection with the circle. Each step logically leads to the conclusion, demonstrating the relationship between the curve and the circle.