StemCET Logo

Maths Question 13 – JEE-MAIN 2026

← Back
If the point of intersection of the lines x+13=y+a5=z+b+17 and x−21=y−b4=z−2a7 lies on xy-plane, then the value of a+b is:

Represent the coordinates of a general point on each line using a parameter.

● Video Walkthrough
Step 1: Represent points on lines and apply z=0 condition✦ Active

Let the first line be equal to λ and the second line be equal to μ. The general point on the first line is P1(3λ−1,5λ−a,7λ−b−1). The general point on the second line is P2(μ+2,4μ+b,7μ+2a). Since the point of intersection lies on the xy-plane, its z-coordinate must be 0.

7λ−b−1=0⇒b=7λ−1(1) 7μ+2a=0⇒a=−72μ(2)
Step 2: Equate coordinates and solve for parameters○ Expand

For the lines to intersect, the coordinates of P1 and P2 must be equal. Equating the x and y coordinates:

3λ−1=μ+2⇒3λ−μ=3(3) 5λ−a=4μ+b(4)

Substitute equations (1) and (2) into equation (4):

5λ−(−72μ)=4μ+(7λ−1) 10λ+7μ=8μ+14λ−2 −4λ−μ=−2⇒4λ+μ=2(5)

Now, solve the system of equations (3) and (5):

(3λ−μ)+(4λ+μ)=3+2⇒7λ=5⇒λ=57 3(57)−μ=3⇒157−μ=3⇒μ=157−3=−67
💡 Teacher's Secret Hint

Ensure careful algebraic manipulation when substituting and solving the system of equations.

Step 3: Calculate a, b, and a+b○ Expand

Substitute the values of λ and μ back into equations (1) and (2) to find a and b:

b=7λ−1=7(57)−1=5−1=4 a=−72μ=−72(−67)=3

Finally, calculate a+b:

a+b=3+4=7
💡 Teacher's Secret Hint

Double-check the calculations for a and b to avoid arithmetic errors.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.