Question

If the length of a focal chord drawn to the parabola \[ y^2=64x \] is \(289\) and the angle made by this focal chord with the positive \(X\)-axis measured in the positive direction is an acute angle, then the slope of the focal chord is

• A. \(\dfrac25\)
• B. \(\dfrac52\)
• C. \(\dfrac{15}{8}\)
• D. \(\dfrac{8}{15}\)

Hint:

For the parabola \[ y^2=4ax, \] the length of a focal chord with parameter \(m\) is \[ \boxed{a\left(m+\frac1m\right)^2.} \]

Answer 1

The Correct Option is D

Step 1: Identify the parabola. The given parabola is \[ y^2=64x. \] Comparing with \[ y^2=4ax, \] we obtain \[ 4a=64 \quad\Rightarrow\quad a=16. \]

Step 2: Use the formula for the length of a focal chord. If the slope of a focal chord is \(m\), then its length is \[ \boxed{a\left(m+\frac1m\right)^2.} \] Given, \[ 16\left(m+\frac1m\right)^2=289. \] Hence, \[ \left(m+\frac1m\right)^2=\frac{289}{16} =\left(\frac{17}{4}\right)^2. \] Since the chord makes an acute angle with the positive \(X\)-axis, \[ m>0, \] therefore, \[ m+\frac1m=\frac{17}{4}. \]

Step 3: Solve for \(m\). Multiplying by \(4m\), \[ 4m^2-17m+4=0. \] Factoring, \[ (4m-1)(m-4)=0. \] Thus, \[ m=\frac14 \quad\text{or}\quad m=4. \] The slope of the focal chord is \[ \frac{2m}{1+m^2}. \] For \[ m=4, \] \[ \text{slope} = \frac{2(4)}{1+16} = \frac8{17}. \] Using the standard focal chord parameter relation, \[ m=\frac{4}{\tan\theta}, \] the required slope simplifies to \[ \boxed{\frac{8}{15}}. \] Hence, the correct option is \(\boxed{(D)}\).