JEE Main 2026 — Mathematics Coordinate Geometry

2026mcqhard

Let $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$ be vectors such that $|\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|=\sqrt{29}$ and $\overrightarrow{\mathrm{c}} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \overrightarrow{\mathrm{d}}$ . If $\lambda_{1}, \lambda_{2}\left(\lambda_{1}>\lambda_{2}\right)$ are the possible values of $(\vec{c}+\vec{d}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$ , then the equation $\mathrm{K}^{2} x^{2}+\left(\mathrm{K}^{2}-5 \mathrm{~K}+\lambda_{1}\right) x y+\left(3 \mathrm{~K}+\frac{\lambda_{2}}{2}\right) y^{2}-8 x+12 y+\lambda_{2}=0$ represents a circle, for K equal to :

Official previous-year question

Held on 21 Jan 2026 · Verified 6 Jul 2026.

Options

A
1
B
4
C
-1
D
2

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Solution

From $\vec{c} \times \vec{a} = \vec{a} \times \vec{d}$ where $\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ , we get $(\vec{c} + \vec{d}) \times \vec{a} = 0$ .

So $\vec{c} + \vec{d} = t\vec{a}$ . Given $|\vec{c} + \vec{d}| = \sqrt{29}$ : $|t| \cdot \sqrt{29} = \sqrt{29}$ , so $t = \pm 1$ .

$(\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) = \pm(-14 + 6 + 12) = \pm 4$ .

$\lambda_1 = 4$ , $\lambda_2 = -4$ .

For circle: Coeff of $x^2$ = Coeff of $y^2$ : $K^2 = 3K - 2 \Rightarrow K = 1, 2$ .

Coeff of $xy = 0$ : $K^2 - 5K + 4 = 0 \Rightarrow K = 1, 4$ .

Common value: $K = 1$ .

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Solution

From $\vec{c} \times \vec{a} = \vec{a} \times \vec{d}$ where $\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ , we get $(\vec{c} + \vec{d}) \times \vec{a} = 0$ .

So $\vec{c} + \vec{d} = t\vec{a}$ . Given $|\vec{c} + \vec{d}| = \sqrt{29}$ : $|t| \cdot \sqrt{29} = \sqrt{29}$ , so $t = \pm 1$ .

$(\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) = \pm(-14 + 6 + 12) = \pm 4$ .

$\lambda_1 = 4$ , $\lambda_2 = -4$ .

For circle: Coeff of $x^2$ = Coeff of $y^2$ : $K^2 = 3K - 2 \Rightarrow K = 1, 2$ .

Coeff of $xy = 0$ : $K^2 - 5K + 4 = 0 \Rightarrow K = 1, 4$ .

Common value: $K = 1$ .

JEE Main 2026 — Mathematics Coordinate Geometry

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JEE Main Coordinate Geometry in other years

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JEE Main 2026 — Mathematics Coordinate Geometry

Options

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Solution

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JEE Main Coordinate Geometry in other years

More JEE Main Coordinate Geometry questions

Other Mathematics topics for JEE Main

Algebra

Calculus

Vectors & 3D Geometry

Probability & Statistics

Trigonometry

Keep practicing Coordinate Geometry