Question

The angle between the curves $y^2=4x$ and $x^2=2y-3$ at the point $(1,2)$ is

• A. ${60}^{\circ }$
• B. ${30}^{\circ }$
• C. ${90}^{\circ }$
• D. $0^{\circ }$

Answer 1

The correct option is D $0^{\circ }$

Let the slope of the tangent to the curves at the point of intersection be $m_1,m_2$

Slope of tangent for $y^2=4x$ at $(1,2)$ is
$2y\frac{dy}{dx}=4$
$\Rightarrow \frac{dy}{dx}=\frac{2}{y}$
$=m_1=1\cdots \left(i\right)$
Similary, slope of tangent to $x^2=2y-3$
$2x=2\frac{dy}{dx}$
$\Rightarrow \frac{dy}{dx}=x$
​​​​​$\Rightarrow m_2=1$
$\therefore m_1=m_2=1$
$\Rightarrow$ slope of the tangents are equal and both passes through the point $(1,2)$ ,Hence the tangents coincide
$\Rightarrow$ Angle between them is $0^{\circ }$​