SATHEE: Circle 1 Question 17

Circle 1 Question 17

17. Let $C$ be any circle with centre $(0, \sqrt{2})$ . Prove that at most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

(1997, 5M)

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Answer:

Correct Answer: 17. (a)

Solution:

Equations of any circle $C$ with centre at $(0, \sqrt{2})$ is given by

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

where, $r > 0$ .

Let $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ be three distinct rational points on circle. Since, a straight line parallel to $X$ -axis meets a circle in at most two points, either $y_{1}, y_{2}$ or $y_{1}, y_{3}$ .

On putting these in Eq. (i), we get

$\begin{aligned} x_{1}^{2} + y_{1}^{2} - 2 \sqrt{2} y_{1} = r^{2} - 2 \\ x_{2}^{2} + y_{2}^{2} - 2 \sqrt{2} y_{2} = r^{2} - 2 \\ x_{3}^{2} + y_{3}^{2} - 2 \sqrt{2} y_{3} = r^{2} - 2 \end{aligned}$

On subtracting Eq. (ii) from Eq. (iii), we get

$where, \begin{aligned} p_{1} - \sqrt{2} q_{1} & = 0 \\ p_{1} & = x_{2}^{2} + y_{2}^{2} - x_{1}^{2} - y_{1}^{2}, \\ q_{1} & = y_{2} - y_{1} \end{aligned}$

On subtracting Eq. (ii) from Eq. (iv), we get

$where \begin{aligned} p_{2} - \sqrt{2} q_{2} & = 0 \\ p_{2} & = x_{3}^{2} + y_{3}^{2} - x_{1}^{2} - y_{1}^{2}, q_{2} = y_{3} - y_{1} \end{aligned}$

Now, $p_{1}, p_{2}, q_{1}, q_{2}$ are rational numbers. Also, either $q_{1} \neq 0$ or $q_{2} \neq 0$ . If $q_{1} \neq 0$ , then $\sqrt{2} = p_{1} / q_{1}$ and if $q_{2} \neq 0$ , then $\sqrt{2} = p_{2} / q_{2}$ . In any case $\sqrt{2}$ is a rational number. This is a contradiction.

Circle 1 Question 17

17. Let $C$ be any circle with centre $(0, \sqrt{2})$ . Prove that at most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

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Circle 1 Question 17

17. Let C be any circle with centre (0,2). Prove that at most two rational points can be there on C. (A rational point is a point both of whose coordinates are rational numbers.)

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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17. Let $C$ be any circle with centre $(0, \sqrt{2})$ . Prove that at most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)