SATHEE: Parabola 3 Question 1

Parabola 3 Question 1

1. If the parabolas $y^{2} = 4 b (x - c)$ and $y^{2} = 8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a, b, c)$ ?

(2019 Main, 10 Jan, I)

(a) $\frac{1}{2}, 2, 0$

(b) $(1, 1, 0)$

(d) $\frac{1}{2}, 2, 3$

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Answer:

Correct Answer: 1. (c)

Solution:

Key Idea (i) First find the focus of the given parabola (ii) Then, find the slope of the focal chord by using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

(iii) Now, find the length of the focal chord by using the formula $4 a {cosec}^{2} α$ .

Equation of given parabola is $y^{2} = 16 x$ , its focus is $(4, 0)$ . Since, slope of line passing through $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by $m = \tan θ = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$∴$ Slope of focal chord having one end point is $(1, 4)$ is

$m = \tan α = \frac{4 - 0}{1 - 4} = - \frac{4}{3}$

[where, ’ $α$ ’ is the inclination of focal chord with $X$ -axis.] Since, the length of focal chord $= 4 a {cosec}^{2} α$

$∴$ The required length of the focal chord

$\begin{aligned} = 16 [1 + \cot^{2} α] [∵ a = 4 and {cosec}^{2} α = 1 + \cot^{2} α] \\ = 161 + \frac{9}{16} = 25 units ∵ \cot α = \frac{1}{\tan α} = - \frac{3}{4} \end{aligned}$

Parabola 3 Question 1

1. If the parabolas $y^{2} = 4 b (x - c)$ and $y^{2} = 8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a, b, c)$ ?

Answer:

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Parabola 3 Question 1

1. If the parabolas y2=4b(x−c) and y2=8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a,b,c) ?

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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1. If the parabolas $y^{2} = 4 b (x - c)$ and $y^{2} = 8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a, b, c)$ ?