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

Answer: (C)

$2x - y + 1 = 0$

Answer 2

Given equation of parabola is
$y=x^{2}-2 x+5\,...(i)$
By putting $x_{1}=1, x_{2}=3$ in E (i), we get
$y_{1}=1$ and $y_{2}=8$
$\therefore$ 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
$y-4=\frac{8-4}{3-1}(x-1)$
$y-4=2 x-2$
$\Rightarrow \, 2 x-y+2=0$
Now, equation of tangent parallel to chord will be
$2 x-y+k=0\,...(ii)$
In given options, only option (b) satisfies the condition for E (iii)
i.e. $2 x-y+1=0\,...(iii)$