PRACTICE PROBLEM AS PER CBSE COORDINATE GEOMETRY

Great! Let’s take another problem and format it according to the CBSE examination style. Here’s an example for Practice Problem 8: Intersection of Two Lines:

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Question

Find the point of intersection of the lines x+2y=4x + 2y = 4 and 3x−y=53x – y = 5.

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Solution

Step 1: Write the given equations

The given equations are:

(1)x+2y=4(1) \quad x + 2y = 4 (2)3x−y=5.(2) \quad 3x – y = 5.

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Step 2: Solve the equations simultaneously

From equation (1):

x=4−2y.(3)x = 4 – 2y \tag{3}.

Substitute x=4−2yx = 4 – 2y into equation (2):

3(4−2y)−y=5.3(4 – 2y) – y = 5.

Simplify:

12−6y−y=5.12 – 6y – y = 5. 12−7y=5⇒7y=12−5=7.12 – 7y = 5 \quad \Rightarrow \quad 7y = 12 – 5 = 7. y=1.y = 1.

Substitute y=1y = 1 into equation (3):

x=4−2(1)=4−2=2.x = 4 – 2(1) = 4 – 2 = 2.

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Step 3: Write the solution

The point of intersection is:

(2,1).(2, 1).

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Graphical Representation

The graph below shows the lines x+2y=4x + 2y = 4 and 3x−y=53x – y = 5, intersecting at the point (2,1)(2, 1).

Graph Explanation

• The blue line represents the equation x+2y=4x + 2y = 4.
• The orange line represents the equation 3x−y=53x – y = 5.
• The lines intersect at the point (2,1)(2, 1), which is highlighted in red.
• Let’s tackle another problem formatted for the CBSE Class 9 Mathematics examination pattern. Here’s a new example:
  

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  Question

  Draw the graph of the linear equation 2x+3y=122x + 3y = 12. Use the graph to find:

  1. The xx– and yy-intercepts.
  2. The coordinates of a point where y=2y = 2.

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  Solution

  Step 1: Rewrite the equation

  The given equation is:

  2x+3y=12.2x + 3y = 12.Rewriting in slope-intercept form (y=mx+cy = mx + c):

  3y=−2x+12⇒y=−23x+4.3y = -2x + 12 \quad \Rightarrow \quad y = -\frac{2}{3}x + 4.

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  Step 2: Find intercepts

  • xx-Intercept: Set y=0y = 0:2x+3(0)=12⇒x=122=6.2x + 3(0) = 12 \quad \Rightarrow \quad x = \frac{12}{2} = 6.So, (6,0)(6, 0).
  • yy-Intercept: Set x=0x = 0:2(0)+3y=12⇒y=123=4.2(0) + 3y = 12 \quad \Rightarrow \quad y = \frac{12}{3} = 4.So, (0,4)(0, 4).

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  Step 3: Find coordinates for y=2y = 2

  Substitute y=2y = 2 into the equation 2x+3y=122x + 3y = 12:

  2x+3(2)=12⇒2x+6=12⇒2x=6⇒x=3.2x + 3(2) = 12 \quad \Rightarrow \quad 2x + 6 = 12 \quad \Rightarrow \quad 2x = 6 \quad \Rightarrow \quad x = 3.So, the point is (3,2)(3, 2).

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  Step 4: Plot the graph

  Plot the points (6,0)(6, 0), (0,4)(0, 4), and (3,2)(3, 2) on the graph and draw the line representing 2x+3y=122x + 3y = 12.

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  Graphical Representation

  Let’s visualize the graph of the equation.

  Graph Explanation

  • The blue line represents the equation 2x+3y=122x + 3y = 12.
  • The red point is the xx-intercept (6,0)(6, 0), where the line crosses the x-axis.
  • The green point is the yy-intercept (0,4)(0, 4), where the line crosses the y-axis.
  • The purple point is (3,2)(3, 2), the solution for y=2y = 2.

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  Final Answer

  1. Intercepts:
     • xx-Intercept: (6,0)(6, 0).
     • yy-Intercept: (0,4)(0, 4).
  2. Point where y=2y = 2: (3,2)(3, 2).

  Would you like another example or a different topic?

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This approach ensures clarity and aligns with the CBSE marking pattern, focusing on:

1. Stepwise solution of simultaneous equations.
2. Verification through graphical representation.
3. Proper annotations and labeling.

Would you like to explore more such problems, or focus on a specific topic?