Parabola

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Question

$Line y = 2 x - b cuts the parabola y = x^{2} - 4 x at points A and B . The value of b for which the ∠ A O B is a right angle is$

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Solution

$\begin{aligned} Line y = 2 x - b \\ \Rightarrow 1 = \frac{2 x - y}{b} \end{aligned}$
$\begin{aligned} Homogenizing parabola with line \\ x^{2} - 4 x (\frac{2 x - y}{b}) - y (\frac{2 x - y}{b}) = 0 \end{aligned}$
$\begin{array}{l} Since ∠ A O B = 90^{\circ} \\ ∴ coefficient of x^{2} + coefficient of y^{2} = 0 \end{array}$
$\begin{array}{l} \Rightarrow 1 - \frac{8}{b} + \frac{1}{b} = 0 \\ \Rightarrow b = 7 \end{array}$

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Parabola

Remember concepts with our Masterclasses.

Line y=2x−b cuts the parabola y=x2−4x at points A and B . The value of b for which the ∠AOB is a right angle is

Line y=2x−b⇒ 1=2x−yb Homogenizing parabola with line x2−4x2x−yb−y2x−yb=0 Since ∠AOB=90∘∴ coefficient of x2+ coefficient of y2=0⇒ 1−8b+1b=0⇒ b=7

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$Line y = 2 x - b cuts the parabola y = x^{2} - 4 x at points A and B . The value of b for which the ∠ A O B is a right angle is$