Prove that the normal chord to a parabola at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.

To prove that the normal chord to a parabola at the point where the ordinate equals the abscissa subtends a right angle at the focus, we will consider the standard equation of a parabola and the properties of its normal chord.

Setting Up the Problem

Let’s take the standard form of the parabola: y² = 4ax. The focus of this parabola is at the point (a, 0).

Identifying the Point on the Parabola

We need to find the point on the parabola where the ordinate (y) is equal to the abscissa (x). Thus, we set y = x. Substituting this into the parabola's equation gives:

x² = 4ax

This simplifies to:

x(x - 4a) = 0

From this, we find two points: (0, 0) and (4a, 4a).

Finding the Normal at the Point (4a, 4a)

Next, we will derive the equation of the normal at the point (4a, 4a). The slope of the tangent at this point can be found using the derivative:

dy/dx = 2y/(4a) = y/(2a)

At (4a, 4a), the slope of the tangent is:

m_t = 4a/(2a) = 2

The slope of the normal, being the negative reciprocal, is:

m_n = -1/2

Equation of the Normal Line

The equation of the normal line at (4a, 4a) can be expressed as:

y - 4a = -1/2(x - 4a)

Rearranging gives:

y = -1/2x + 6a

Finding the Intersection with the Parabola

To find where this normal intersects the parabola again, substitute y = -1/2x + 6a into the parabola's equation:

(-1/2x + 6a)² = 4ax

Expanding and simplifying leads to a quadratic equation in terms of x. Solving this will yield the x-coordinates of the intersection points.

Verifying the Right Angle at the Focus

To show that the normal chord subtends a right angle at the focus, we need to find the slopes of the segments from the focus (a, 0) to the points of intersection. If the product of these slopes equals -1, then the angle is right.

Calculating the slopes and confirming their product will complete the proof.

Conclusion

Through this process, we can demonstrate that the normal chord at the point where the ordinate equals the abscissa indeed subtends a right angle at the focus of the parabola. This illustrates the unique geometric properties of parabolas and their normals.