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

step 1 Hello everyone from the question we are given that if the tangent at point P 1 .1 on curve y squ

If the tangent at P(1,1) on y^2=x(2-x)^2 meets the curve...

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Key Concepts

Implicit Differentiation

This technique is used to differentiate equations that define y implicitly in terms of x. It allows one to obtain the derivative dy/dx even when y is not isolated, which is essential for finding slopes of tangents to curves defined by such equations.

Tangent Line to a Curve

The tangent line at a point on a curve is the straight line that best approximates the curve at that point. It is determined by the point of tangency and the slope of the curve at that point, which is obtained via differentiation.

Intersection of Curves

Finding where a tangent line meets its original curve involves solving a system of equations. This concept is important in analytic geometry for determining multiple points of contact between a line and a curve, including tangency and other intersections.

Solving Polynomial Equations

When substituting the tangent line into the curve's equation, one often obtains a polynomial equation. Solving this polynomial, typically through factorization or other algebraic techniques, reveals the intersection points and identifies the required solutions.

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