SATHEE: Circle 5 Question 16

Circle 5 Question 16

16. Let $2 x^{2} + y^{2} - 3 x y = 0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius 3 with centre in the first quadrant. If $A$ is one of the points of contact, find the length of $O A$ .

$(2001, 5$ M)

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Answer:

Correct Answer: 16. $3 (3 + \sqrt{10})$

Solution:

$2 x^{2} + y^{2} - 3 x y = 0$

[given]

$\begin{aligned} \Rightarrow & 2 x^{2} - 2 x y - x y + y^{2} & = 0 \\ \Rightarrow & 2 x (x - y) - y (x - y) & = 0 \\ \Rightarrow & (2 x - y) (x - y) & = 0 \\ \Rightarrow & y & = 2 x, y = x \end{aligned}$

are the equations of straight lines passing through origin.

Now, let the angle between the lines be $2 θ$ and the line $y = x$ makes angle of $45^{\circ}$ with $X$ -axis.

Therefore, $\tan (45^{\circ} + 2 θ) = 2$ (slope of the line $y = 2 x)$

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$\begin{aligned} \overset{x_{1}}{\to} \\ \Rightarrow \frac{\tan 45^{\circ} + \tan 2 θ}{1 - \tan 45^{\circ} \times \tan 2 θ} = 2 \Rightarrow \frac{1 + \tan 2 θ}{1 - \tan 2 θ} = 2 \\ \Rightarrow \frac{(1 + \tan 2 θ) - (1 - \tan 2 θ)}{(1 + \tan 2 θ) + (1 - \tan 2 θ)} = \frac{2 - 1}{(2 + 1)} = \frac{1}{3} \\ \Rightarrow \frac{2 \tan 2 θ}{2} = \frac{1}{3} \Rightarrow \tan 2 θ = \frac{1}{3} \\ \Rightarrow \frac{2 \tan θ}{1 - \tan^{2} θ} = \frac{1}{3} \\ \Rightarrow (2 \tan θ) \cdot 3 = 1 - \tan^{2} θ \\ \Rightarrow \tan^{2} θ + 6 \tan θ - 1 = 0 \\ \Rightarrow \tan θ = \frac{- 6 \pm \sqrt{36 + 4 \times 1 \times 1}}{2} = \frac{- 6 \pm \sqrt{40}}{2} \\ \Rightarrow \tan θ = - 3 \pm \sqrt{10} \\ \Rightarrow \tan θ = - 3 + \sqrt{10} ∵ 0 < θ < \frac{π}{4} \\ \tan θ = \frac{3}{O A} \\ ∴ O A = \frac{3}{\tan θ} = \frac{3}{(- 3 + \sqrt{10})} = \frac{3 (3 + \sqrt{10})}{(- 3 + \sqrt{10}) (3 + \sqrt{10})} \\ = \frac{3 (3 + \sqrt{10})}{(10 - 9)} = 3 (3 + \sqrt{10}) \end{aligned}$

Circle 5 Question 16

16. Let $2 x^{2} + y^{2} - 3 x y = 0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius 3 with centre in the first quadrant. If $A$ is one of the points of contact, find the length of $O A$ .

Answer:

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Circle 5 Question 16

16. Let 2x2+y2−3xy=0 be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

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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16. Let $2 x^{2} + y^{2} - 3 x y = 0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius 3 with centre in the first quadrant. If $A$ is one of the points of contact, find the length of $O A$ .