Question

The curve whose tangent at any point passes through $(1,2)$ and has a slope that is half that of the line from the origin to the point is a ?

• A. Rectangle hyperbola
• B. Circle
• C. Parabola
• D. st. line through the origin.

Answer 1

The correct option is D st. line through the origin.

Solution:

$m=\frac{y_2-y_1}{x_2-x_1}$

Let $(x_1,y_1)=(0,0)$

$(x_2,y_2)=(1,2)$

Since slope is $m$ which is equal to $\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{1}{2}m$

$\frac{dy}{dx}=\frac{1}{2}\left(2\right)$

$dy=dx$

Integrate on both sides

$\int dy=\int dx$

$y=x+c$ If $c=0$ (constant we can take that)

then $y=x$

which is a straight line pass through it

Answer D