The Intersection of the Parabola ( y^2 x ) and the Line ( y xc )
Understanding the relationship between the parabolic curve ( y^2 x ) and the line ( y xc ) involves exploring their tangents and normals. In this article, we will delve into the specifics of how these two mathematical entities interact and the conditions under which ( y xc ) can be a normal to the parabola ( y^2 x ).
Slope of the Parabola ( y^2 x )
The equation of the parabola is given as ( y^2 x ). To find the slope of the tangent to this parabola at any point ((x, y)), we differentiate implicitly:
Starting with ( y^2 x ), we differentiate both sides with respect to ( x ):
[ 2y frac{dy}{dx} 1 ]
From this, we can solve for the slope ( frac{dy}{dx} ):
[ frac{dy}{dx} frac{1}{2y} ]
Condition for Normal Line
A normal line is perpendicular to the tangent line at the point of intersection. If we want the line ( y xc ) to be a normal to the parabola ( y^2 x ), it must be perpendicular to the tangent at the corresponding point.
The slope of the line ( y xc ) is ( c ). For it to be perpendicular to the tangent, we set its slope ( c ) equal to the negative reciprocal of the tangent slope ( frac{1}{2y} ):
[ c -frac{1}{frac{1}{2y}} -2y ]
At the point of intersection, the slope of the normal must be (-2y) and should be equal to (-1) (since the tangent slope is 1 for the normal to be perpendicular):
[ -2y -1 ]
From this, we find:
[ y frac{1}{2} ]
Substituting ( y frac{1}{2} ) into the parabolic equation ( y^2 x ):
[ left(frac{1}{2}right)^2 x ]
Thus, we get:
[ x frac{1}{4} ]
The point of intersection is ( left(frac{1}{4}, frac{1}{2}right) ).
Equation of the Normal Line
To find the equation of the normal line, we use the point-slope form of the line equation:
The slope of the normal is ( c -1 ) (since ( c -2 cdot frac{1}{2} )), and it passes through the point ( left(frac{1}{4}, frac{1}{2}right) ).
Using the point-slope form ( y - y_1 m(x - x_1) ), we get:
[ y - frac{1}{2} -1 left(x - frac{1}{4}right) ]
Simplifying, we obtain the equation of the normal:
[ y - frac{1}{2} -x frac{1}{4} ]
[ y -x frac{1}{4} frac{1}{2} ]
[ y -x frac{3}{4} ]
The slope ( c ) of the line ( y xc ) is (-frac{3}{4}), meaning the line ( y -frac{3}{4}x ) is the normal to the parabola ( y^2 x ) at the intersection point ( left(frac{1}{4}, frac{1}{2}right) ).
Conclusion
Through this exploration, we have discovered the specific conditions under which the line ( y xc ) can be a normal to the parabola ( y^2 x ). The analysis involves implicit differentiation and the application of basic calculus principles, confirming that only at the single point ( left(frac{1}{4}, frac{1}{2}right) ) does this relationship hold true. Understanding these concepts deepens our knowledge of the geometric properties of parabolas and lines.