Class X Session 2023-24 Question 7

MATH SAMPLE QUESTION PAPER

Class X Session 2023-24

MATHEMATICS STANDARD (Code No.041) 

Question 7 

A point (x,y) is at a distance of 5 units from the origin. How many such points lie in the third

quadrant?

(a) 0 

(b) 1 

(c) 2 

(d) infinitely many

Explanation : 

Distance between points P(x1,y1) and 𝑄(x2,y2) .

As we know that formula of distance between points P and 𝑄 will be 

\[\sqrt{(x2 – x1)^2 + (y2 – y1)^2} \]

Given that the point is at a distance of 5 units from the origin (0,0) we have

\[\Rightarrow  \sqrt{(x-0)^2 + (y-0)^2} = \; 5,-----(a) \]

\[OR \; \Rightarrow  \sqrt{(x)^2 + (y)^2} = \; 5 \]

Squaring both sides to eliminate the square root

\[ OR \; \Rightarrow \ x^2 \; + \;y^2 \; = \; 5^2\]

ah! this equation represents a circle with radius 5.

As we know in the third quadrant, both x and y are negative(-,-)

 If we substitute 𝑥 = −𝑎 and 𝑦 = −𝑏 into the equation, where a and b are positive values

\[\therefore \ a^2 \; + \;b^2  \; = \ 5^2\]

This is the same equation as the one representing the circle, so any point on the circle in the third quadrant will satisfy the given conditions.

Therefore, there are infinitely many points on the circle in the third quadrant that are at a distance of 5 units from the origin

Guess the Option and comment below  👇