Class X Session 2023-24 Question 6

MATH SAMPLE QUESTION PAPER

Class X Session 2023-24

MATHEMATICS STANDARD (Code No.041)

Question 6

What is the ratio in which the line segment joining (2,-3) and (5, 6) is divided by x-axis?

(a) 1:2

 (b) 2:1 

(c) 2:5 

(d) 5:2

Explanation : 

Lets understand first the equation of the line passing through two points (x1,y1) and (x2,y2).

\[\Rightarrow  (y-y1) = \frac{y2-y1}{x2-x1}\times (x-x1), -------(a) \]

lets consider  two points 𝑃 and 𝑄 are arbitrary 

\[ \therefore 𝑃(2,−3) \; and \; 𝑄(5,6) \]

similar to \[ 𝑃(x1, y1) \; and \; 𝑄(x2,y2) \]

Now equation (a) becomes 

\[\Rightarrow  (y-(-3) = \frac{6-(-3)}{5-2}\times (x-2), -------(b) \]

\[ \therefore (y+3) = \frac{6+3}{5-2}\times (x-2) \]

\[\therefore (y+3) = \frac{9}{3}\times (x-2) \]

\[\therefore (y+3) = {3}\times (x-2) \]

\[\therefore (y+3) = {3x-6} \]

\[\; or \; y= {3x-6 -3} \]

\[\; or \; y= {3x-9} ,----(c) \]

To find the y-coordinate of the point where the line intersects the x-axis, we substitute y=0 into the the eq (c)

\[\therefore 0= {3x-9} \]

\[\therefore x = {3} \]

So, the line intersects the x-axis at the point (3,0)

According to given equation we find the lengths of the segments formed by this intersection point and the given points

Length of segment 𝑃𝑄 = Distance between points P and 𝑄 .

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

\[\sqrt{(x2 – x1)^2 + (y2 – y1)^2} \]

\[ PQ \ = \sqrt{(5 – 2)^2 + (6 – (-3))^2} \]

\[\Rightarrow PQ \ = \sqrt{(3)^2 + (9)^2} \]

\[\Rightarrow PQ \ = \sqrt{90} \]

Now , length of segment 𝑃𝑋 = Distance between points 𝑃 and X (intersection point) 

\[ P𝑋 \ = \sqrt{(3 – 2)^2 + (0 – (-3))^2} \]

\[\Rightarrow P𝑋 \ = \sqrt{(1)^2 + (3)^2} \]

\[\Rightarrow P𝑋 \ = \sqrt{10} \]

Now,length of segment 𝑄𝑋 = Distance between points Q and 𝑋 (intersection point):

\[ Q𝑋 \ = \sqrt{(5 – 3)^2 + (0 – (6))^2} \]

\[\Rightarrow Q𝑋 \ = \sqrt{(2)^2 + (6)^2} \]

\[\Rightarrow Q𝑋 \ = \sqrt{40} \]

Now, to find the ratio in which the line segment is divided by the x-axis, we use the formula

\[Ratio\; =\; \frac{Length \; of \; segment \; P𝑋}{Length \; of \; segment \; Q𝑋}\]

\[Ratio\; =\; \frac{\sqrt{10}}{\sqrt{40}}\] 

\[Ratio\; =\; \frac{\sqrt{10}}{{2}\sqrt{10}}\]

\[Ratio\; =\; \frac{1}{2}\] 

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