If the tangent at a variable point P on the curve y=x^2-x^3 meets it again at Q. Show that the l

Name: Problem 53, Chapter 8: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2021-07-02T00:46:21-08:00
Duration: 1 min 46 s
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Description: If the tangent at a variable point P on the curve y=x^2-x^3 meets it again at Q. Show that the locus of the middle point of P Q is y=1-9 x+28 x^2-28 x^3.

step 1 So this question wants us to find other points that have the same derivative as at this point on

If the tangent at a variable point P on the curve y=x^2-x^3...

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Key Concepts

Tangent Line

In differential calculus and analytic geometry, the tangent line to a curve at a point is the line that just ‘touches’ the curve at that point. Its slope is given by the derivative of the curve's function at that point, and it provides the best linear approximation of the curve near the point of contact.

Intersection of Curves

This concept involves finding the points where two curves or a curve and a line intersect by solving their equations simultaneously. In problems where a tangent meets the curve again, one typically sets up an equation that captures both the curve and the tangent line and then identifies the common roots which represent the points of intersection.

Midpoint Formula

A key tool in analytic geometry, the midpoint formula calculates the center point between two given points. It is used to determine the coordinates of the middle point by averaging the x-coordinates and y-coordinates separately, which is essential in problems that ask for the locus of such midpoints.

Locus

The locus of a point is the set of all positions that the point can occupy, given certain conditions or constraints. In this context, the problem involves deriving an equation that describes all possible positions of the midpoint of a segment joining two points on a curve, as one of the points moves along the curve.

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