Question

A chord of the parabola $y=x^2-2x+5$ joins the point with the abscissas $x_1=1,x_2=3$. Then the equation of the tangent to the parabola parallel to the chord is :

• A. $2x-y+\frac{5}{4}=0$
• B. $2x-y+2=0$
• C. $2x-y+1=0$
• D. $2x+y+1=0$

Answer 1

The correct option is C $2x-y+1=0$


Coordinates of $A$ and $B,$
$y=1-2+5=4\Rightarrow A\equiv (1,4)$
$y=9-6+5=8\Rightarrow B\equiv (3,8)$

Slope of the chord joining $A$ and $B$ is,
$m=\frac{8-4}{3-1}=2$

Slope of the tangent,
$\frac{dy}{dx}=2x-2=2\Rightarrow x=2$
So, coordinates of $P$ is $(2,5)$
Equation of the tangent that is parallel to the chord,
$y=2x+c\Rightarrow 5=4+c\Rightarrow c=1$

Hence, the equation of the tangent is,
$y=2x+1\Rightarrow 2x-y+1=0$