If the point of intersection of the lines (x+1)/3 = (y+a)/5 = (z+b+1)/7 and (x-2)/1 = (y-b)/4 = (z-2a)/7 lies on xy-plane, then the value of a+b is:

The correct answer is Option 3. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q13 – JEE-MAIN 2026

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Home Maths 3D GeometryQuestion #5

If the point of intersection of the lines

\frac{x + 1}{3} = \frac{y + a}{5} = \frac{z + b + 1}{7}

and

\frac{x - 2}{1} = \frac{y - b}{4} = \frac{z - 2 a}{7}

lies on xy-plane, then the value of

a + b

is:

Clue 1 of 3: Parametric Form of Lines

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

🧠 Full Solution Path

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 $P_{1} (3 λ - 1, 5 λ - a, 7 λ - b - 1)$ . The general point on the second line is $P_{2} (μ + 2, 4 μ + b, 7 μ + 2 a)$ . Since the point of intersection lies on the $x y$ -plane, its $z$ -coordinate must be 0.

7 λ - b - 1 = 0 \Rightarrow b = 7 λ - 1 (1)

7 μ + 2 a = 0 \Rightarrow a = - \frac{7}{2} μ (2)

Step 2: Equate coordinates and solve for parameters○ Expand

For the lines to intersect, the coordinates of $P_{1}$ and $P_{2}$ must be equal. Equating the $x$ and $y$ coordinates:

3 λ - 1 = μ + 2 \Rightarrow 3 λ - μ = 3 (3)

5 λ - a = 4 μ + b (4)

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

5 λ - (- \frac{7}{2} μ) = 4 μ + (7 λ - 1)

10 λ + 7 μ = 8 μ + 14 λ - 2

- 4 λ - μ = - 2 \Rightarrow 4 λ + μ = 2 (5)

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

(3 λ - μ) + (4 λ + μ) = 3 + 2 \Rightarrow 7 λ = 5 \Rightarrow λ = \frac{5}{7}

3 (\frac{5}{7}) - μ = 3 \Rightarrow \frac{15}{7} - μ = 3 \Rightarrow μ = \frac{15}{7} - 3 = - \frac{6}{7}

💡 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 = 7 λ - 1 = 7 (\frac{5}{7}) - 1 = 5 - 1 = 4

a = - \frac{7}{2} μ = - \frac{7}{2} (- \frac{6}{7}) = 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.

Maths Question 13 – JEE-MAIN 2026

Clue 1 of 3: Parametric Form of Lines

🧠 Full Solution Path