SATHEE: Circle 1 Question 13

Circle 1 Question 13

13. Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line $x + y = 1$ . If the intercepts made by the circle $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then which of the following equation can represent $L_{1}$ ?

(a) $x + y = 0$

(b) $x - y = 0$

(d) $x - 7 y = 0$

$(1999, 1 M)$

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Answer:

Correct Answer: 13. (b, c)

Solution:

Let equation of line $L_{1}$ be $y = m x$ . Intercepts made by $L_{1}$ and $L_{2}$ on the circle will be equal i.e. $L_{1}$ and $L_{2}$ are at the same distance from the centre of the circle;

Centre of the given circle is $(1 / 2, - 3 / 2)$ . Therefore,

$\begin{aligned} \frac{| 1 / 2 - 3 / 2 - 1 |}{\sqrt{1 + 1}} & = | \frac{\frac{m}{2} + \frac{3}{2}}{\sqrt{m^{2} - 1}} | \Rightarrow \frac{2}{\sqrt{2}} = \frac{| m + 3 |}{2 \sqrt{m^{2} + 1}} \\ \Rightarrow & 8 m^{2} + 8 & = m^{2} + 6 m + 9 \\ \Rightarrow & 7 m^{2} - 6 m - 1 & = 0 \Rightarrow (7 m + 1) (m - 1) = 0 \\ m & = - \frac{1}{7}, m = 1 \end{aligned}$

Thus, two chords are $x + 7 y = 0$

$and x - y = 0 .$

Therefore, (b) and (c) are correct answers.

Circle 1 Question 13

13. Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line $x + y = 1$ . If the intercepts made by the circle $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then which of the following equation can represent $L_{1}$ ?

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Answer:

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Circle 1 Question 13

13. Let L1 be a straight line passing through the origin and L2 be the straight line x+y=1. If the intercepts made by the circle x2+y2−x+3y=0 on L1 and L2 are equal, then which of the following equation can represent L1 ?

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Answer:

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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13. Let $L_{1}$ be a straight line passing through the origin and $L_{2}$ be the straight line $x + y = 1$ . If the intercepts made by the circle $x^{2} + y^{2} - x + 3 y = 0$ on $L_{1}$ and $L_{2}$ are equal, then which of the following equation can represent $L_{1}$ ?