Question

If the parabola \[ y^{2}=9x \] cuts the ellipse \[ \frac{x^{2}}{9}+\frac{y^{2}}{b}=1 \] orthogonally, then the length of the latus rectum of the given parabola is

• A. \(2b\)
• B. \(\dfrac{b}{2}\)
• C. \(b\)
• D. \(\dfrac{b}{3}\)

Hint:

For two curves intersecting orthogonally, \[ \boxed{m_1m_2=-1.} \] Also, for the parabola \[ y^{2}=4ax, \] the length of the latus rectum is \[ \boxed{4a.} \]

Answer 1

The Correct Option is B

Step 1: Find the slope of the parabola. The parabola is \[ y^{2}=9x. \] Differentiating, \[ 2y\frac{dy}{dx}=9, \] so \[ m_1=\frac{dy}{dx}=\frac{9}{2y}. \]

Step 2: Find the slope of the ellipse. The ellipse is \[ \frac{x^{2}}{9}+\frac{y^{2}}{b}=1. \] Differentiating implicitly, \[ \frac{2x}{9}+\frac{2y}{b}\frac{dy}{dx}=0, \] hence \[ m_2=\frac{dy}{dx} = -\frac{bx}{9y}. \]

Step 3: Use the orthogonality condition. Since the curves intersect orthogonally, \[ m_1m_2=-1. \] Therefore, \[ \frac{9}{2y}\left(-\frac{bx}{9y}\right)=-1. \] Using the parabola equation \[ y^{2}=9x, \] we get \[ -\frac{bx}{2y^{2}} = -\frac{b}{18} = -1. \] Hence, \[ \boxed{b=18.} \]

Step 4: Find the latus rectum of the parabola. Since \[ y^{2}=9x=4ax, \] we have \[ 4a=9 \quad\Rightarrow\quad a=\frac94. \] The length of the latus rectum is \[ 4a=9. \] Also, \[ \frac{b}{2} = \frac{18}{2} = 9. \] Hence, \[ \boxed{\text{Length of latus rectum}=\frac{b}{2}.} \] Therefore, the correct option is \(\boxed{(B)}\).