Question

If the line y = 4 + kx , k > 0, is the tangent to the parabola y = x – x 2 at the point P and V is the vertex of the parabola, then the slope of the line through P and V is:

• A. $\frac{3}{2}$
• B. $\frac{26}{9}$
• C. $\frac{5}{2}$
• D. $\frac{23}{6}$

Answer 1

Answer: (A)

$\frac{3}{2}$

Answer 2

The correct option is(C): $\frac{5}{2}$.

∵ Line y = kx + 4 touches the parabola y = x – x2.

So, kx + 4 = x – x2 ⇒ x2 + (k – 1) x + 4 = 0 has only one root

(k – 1)2 = 16 ⇒ k = 5 or – 3 but k > 0

So, k = 5.

And hence x2 + 4x + 4 = 0 ⇒ x = – 2

So, P(–2, –6) and V is

$(\frac{1}{2},\frac{1}{4})$

Slope of PV

$=\frac{\frac{1}{4}+6}{\frac{1}{2}+2}=\frac{5}{2}$