Question

If two tangents from point \((h,k)\) to parabola \(y^2 = 64x\) have slopes such that one is 8 times the other, then value of \( \frac{k^2}{2h} \) is:

• A. \(9\)
• B. \(27\)
• C. \(81\)
• D. \(162\)

Hint:

Use tangent slope form for parabola and relation between roots.

Answer 1

The Correct Option is C

Concept: Tangent to parabola \(y^2 = 4ax\): \[ y = mx + \frac{a}{m} \] Here \(a = 16\)

Step 1: Point \((h,k)\) lies on tangent: \[ k = mh + \frac{16}{m} \]

Step 2: Rearrange: \[ mh^2 - kh + 16 = 0 \] Slopes \(m_1, m_2\) satisfy: \[ m_1 m_2 = \frac{16}{h} \]

Step 3: Given \(m_1 = 8m_2\) \[ m_1 m_2 = 8m_2^2 = \frac{16}{h} \]

Step 4: Using sum/product relations: \[ \frac{k^2}{2h} = 81 \]