Question

The angle between the tangents to the curve $y^2=2ax$ at the points where $x=\frac{a}{2}$, is

• A. π/6
• B. π/4
• C. π/3
• D. π/2

Answer 1

The correct option is D

π/2

We have, $y^2=2ax$
Put $x=\frac{a}{2};y^2=2a\left(\frac{a}{2}\right)\Rightarrow y=\pm a$
$\therefore$ The points are $(\frac{a}{2},a)$ and $(\frac{a}{2},-a)$
On differentiating equarion (1) w.r.t.x, we get
$2y\frac{dy}{dx}=2a\Rightarrow \frac{dy}{dx}=\frac{a}{y}$
At $(\frac{a}{2},a)\frac{dy}{dx}=\frac{a}{y}=\frac{a}{a}=1=m_1\left(say\right).$
At $(\frac{a}{2},-a),\frac{dy}{dx}=\frac{a}{y}=\frac{a}{a}=-1=m_2\left(say\right).$
Since $m_1{m}_2=-1$, the two tangents are at right are at right angles.
Hence (d) is the correct answer.