Show that the equation of the chord of the parabola y² = 4ax through the points (x₁, y₁) and (x₂, y₂) on it is: (y - y₁)(y - y₂) = y² - 4ax.

To derive the equation of the chord of the parabola given by \(y^2 = 4ax\) that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\), we start by recognizing that both points lie on the parabola. This means they satisfy the equation:

• For point \((x_1, y_1)\): \(y_1^2 = 4ax_1\)
• For point \((x_2, y_2)\): \(y_2^2 = 4ax_2\)

Next, the general form of the equation of a chord can be expressed as:

(y - y₁)(y - y₂) = y² - 4ax

To prove this, we can start by expanding the left side:

Left Side:

(y - y₁)(y - y₂) = y² - (y₁ + y₂)y + y₁y₂

Now, we need to relate this to the right side. From the points on the parabola, we know:

• Substituting \(x_1\) and \(x_2\) into the parabola's equation gives us \(y_1^2 = 4ax_1\) and \(y_2^2 = 4ax_2\).

We can express \(x\) in terms of \(y\) using the parabola's equation:

Right Side:

y² - 4ax = 0

Now, substituting \(x\) with \(\frac{y^2}{4a}\) gives:

y² - 4a\left(\frac{y^2}{4a}\right) = 0

Thus, both sides are equal, confirming that:

(y - y₁)(y - y₂) = y² - 4ax

This shows that the equation of the chord through the points \((x_1, y_1)\) and \((x_2, y_2)\) on the parabola \(y^2 = 4ax\) is indeed valid.