Question

Equation of a tangent line of the parabola \(y^2 = 8x\), which passes through the point \((1, 3)\) is:

• A. \(y = 2x + 1\)
• B. \(2y = x + 5\)
• C. \(y = -2x + 5\)
• D. \(2y = 3x + 3\)

Hint:

To find the equation of a tangent to a parabola passing through a given point, substitute the point into the general tangent form and solve for the slope. Always check if the obtained value of the slope satisfies the equation.

Answer 1

The Correct Option is A

Step 1: Start with the equation of the parabola \(y^2 = 8x\). 
Step 2: Use the point-slope form of the equation of a tangent to a parabola. The general form of the tangent line to a parabola \(y^2 = 4ax\) is given by: \[ y = mx + \frac{a}{m}. \] For the given parabola \(y^2 = 8x\), we compare it with \(y^2 = 4ax\) and get \(4a = 8\), which gives \(a = 2\). Substituting \(a = 2\) into the tangent formula, we get: \[ y = mx + \frac{2}{m}. \] Step 3: Substitute the point \((1, 3)\) into the equation of the tangent line to find \(m\). We substitute \(x = 1\) and \(y = 3\) into the equation \(y = mx + \frac{2}{m}\): \[ 3 = m \cdot 1 + \frac{2}{m}. \] Multiplying through by \(m\) to clear the denominator: \[ 3m = m^2 + 2. \] Rearranging the equation: \[ m^2 - 3m + 2 = 0. \] Factoring the quadratic equation: \[ (m - 1)(m - 2) = 0. \] Thus, \(m = 1\) or \(m = 2\). 
Step 4: Since \(m = 2\) satisfies the equation, substitute \(m = 2\) back into the tangent equation. We get: \[ y = 2x + \frac{2}{2} = 2x + 1. \]