NIMCET 2023 — Mathematics PYQ

Solution

Step 1: Convert the General Equation to Standard Form

To identify the type of conic and its properties, we complete the square for both $x$ and $y$.

The given equation is:

Group the $x$ terms and $y$ terms:

Complete the square:

• For $x$: $(x^2+2x+1)$

• For $y$: $-4(y^2-2y+1)$

Adjust the right side of the equation accordingly:

Divide the entire equation by $4$ to get the standard form of a hyperbola:

Step 2: Identify Key Parameters

Comparing this with the standard form $\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$:

• Center $(h,k)$: $(-1,1)$

• $a^2=4⟹a=2$

• $b^2=1⟹b=1$

This is a horizontal hyperbola because the $x$ term is positive.

Step 3: Calculate the Eccentricity ($e$)

For a hyperbola:

Step 4: Find the Distance to the Foci ($ae$)

The distance from the center to each focus is $c=ae$:

Step 5: Determine the Coordinates of the Foci

Since the hyperbola is horizontal, the foci lie on the line $y=k$ (the transverse axis). The coordinates are:

Final Answer:

The foci are $(-1\pm \sqrt{5},1)$. (Option B)