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

Answer 1

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

Given equation of parabola is
$y=x^2-2x+\mathrm{5....}\left(i\right)$
By putting $x_1=1,x_2=3$ in Eq (i) we get
$y_1=4;y_2=8$
$\therefore$ Points on the parabola are $(1,4)$ and $(3,8)$
Equation of the chord of given parabola by joining the points $(1,4)$ and $(3,8)$ will be
$y-4=\frac{8-4}{3-1}(x-1)$
$y-4=2x-2$
$\Rightarrow$ $2x-y+2=0$
Now, equation of tangent parallel to chord will be
$2x-y+k=\mathrm{0...}\left(ii\right)$
In given options, only option (b) satisfies the condition from Eq(iii)
i.e., $2x-y+1=\mathrm{0....}\left(iii\right)$