Question

If the tangent at (1,1) on \( y^2 = x(2 - x^2) \) meets the curve again at P, then P is:

• A. (4,4)
• B. (-1,2)
• C. \(\left( \frac{9}{4}, \frac{3}{8} \right)\)
• D. None of these

Hint:

When finding the point where a tangent meets the curve again, solve for the intersection of the tangent line equation and the original curve.

Answer 1

The Correct Option is C

The equation is \( y^2 = x(2 - x^2) \).
First, differentiate both sides: \[ 2y \frac{dy}{dx} = 2 - 3x^2. \] At the point (1,1), substitute \(x = 1\) and \(y = 1\) into the derivative.
We get the slope of the tangent line.
Next, use the point-slope form of the equation of the tangent.
Substitute the point into the equation of the curve to find where the tangent meets the curve again.
This gives \( P = \left( \frac{9}{4}, \frac{3}{8} \right) \).