Ellipse Latus Rectum Formula for JEE

 

❓ Concept Question

How do we use shortest distance between skew lines and ellipse formulas together in 3D Geometry problems?

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🖼 Concept Image

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✍️ Short Concept

This concept is based on:

👉 Symmetric form of line
👉 Vector form of line
👉 Shortest distance between skew lines
👉 Ellipse latus rectum formula.

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🔷 Step 1 — Symmetric Form of Line 💯

General symmetric form:

$$\frac{𝑥-{𝑥}_1}{𝑎}=\frac{𝑦-{𝑦}_1}{𝑏}=\frac{𝑧-{𝑧}_1}{𝑐}$$

Here:

$$(𝑎,𝑏,𝑐)$$

are direction ratios of the line.

These help identify:

👉 Direction vector of the line.

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🔷 Step 2 — Vector Form of Line

Vector equation of line:

$$\overrightarrow{𝑟}=\overrightarrow{𝑎}+𝜆\overrightarrow{𝑏}$$

Where:

• $\overrightarrow{𝑎}$→ position vector of a point on line

• $\overrightarrow{𝑏}$ → direction vector

• $𝜆$ → parameter

This form is useful in shortest distance problems.

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🔷 Step 3 — Shortest Distance Between Two Lines

For skew lines:

$$𝑆𝐷=\frac{\mathrm{\mid }({\overrightarrow{𝑎}}_2-{\overrightarrow{𝑎}}_1)\cdot ({\overrightarrow{𝑏}}_1\times {\overrightarrow{𝑏}}_2)\mathrm{\mid }}{\mathrm{\mid }{\overrightarrow{𝑏}}_1\times {\overrightarrow{𝑏}}_2\mathrm{\mid }}$$

Where:

• ${\overrightarrow{𝑏}}_1,{\overrightarrow{𝑏}}_2$ are direction vectors

• Cross product gives perpendicular direction

This formula is directly used in JEE.

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🔷 Step 4 — Ellipse Standard Equation

Ellipse equation:

$$\frac{{𝑥}^2}{{𝑎}^2}+\frac{{𝑦}^2}{{𝑏}^2}=1$$

Where:

• $𝑎$ → semi-major axis

• $𝑏$ → semi-minor axis

Larger denominator represents major axis.

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🔷 Step 5 — Length of Latus Rectum

For ellipse:

$$\text{Length of latus rectum}=\frac{2{𝑏}^2}{𝑎}$$

This formula is frequently used in JEE problems.

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🔷 Step 6 — JEE Trap Alert 🚨

❌ Direction vector incorrectly identify kar dena

❌ Cross product calculation galat kar dena

❌ Major and minor axis confuse kar lena

❌ Latus rectum formula ulta use kar dena

Remember:

$$\text{Larger denominator}\Rightarrow \text{major axis}$$

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✅ Final Takeaway

For mixed 3D + conic problems:

$$\boxed{{\displaystyle \text{Line form}\to \text{Direction vectors}\to \text{Shortest distance}\to \text{Ellipse formula}}}$$

Always:

$$\text{Check vectors carefully before substitution}$$

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⭐ Golden JEE Insight

If shortest distance formula contains:

$${\overrightarrow{𝑏}}_1\times {\overrightarrow{𝑏}}_2$$

👉 Cross product is compulsory.

For ellipse:

$$\text{Latus rectum}=\frac{2{𝑏}^2}{𝑎}$$

is the direct shortcut.

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📚 Related Topics

• Find Eccentricity Using Ellipse and Hyperbola Foci
• Shortest Distance Between Lines and Ellipse Relation
• Ellipse Parameters Using Function Minimum Value