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 

$$\therefore 𝑃(2,-3)and𝑄(5,6)$$

similar to $$𝑃(x1,y1)and𝑄(x2,y2)$$

Now equation (a) becomes 

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

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

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

$$\therefore (y+3)=3\times (x-2)$$

$$\therefore (y+3)=3x-6$$

$$ory=3x-6-3$$

$$ory=3x-9,----(c)$$

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

$$Ratio=\frac{LengthofsegmentP𝑋}{LengthofsegmentQ𝑋}$$

$$Ratio=\frac{\sqrt{10}}{\sqrt{40}}$$ 

$$Ratio=\frac{\sqrt{10}}{2\sqrt{10}}$$

$$Ratio=\frac{1}{2}$$ 

Guess the Option and comment below  👇