CUET PG 2026 — Mathematics PYQ

Two curves cut at a right angle (orthogonally) if the product of their slopes ($m_1$ and $m_2$) at the point of intersection is $-1$. That is, $m_1\cdot m_2=-1$.

Step 1: Find the point of intersection

Given curves:

1. $x=y^2$

2. $xy=k$

Substitute $x=y^2$ into the second equation:

Then, $x=(k^{1/3}{)}^2=k^{2/3}$.

So, the intersection point is $(k^{2/3},k^{1/3})$.

Step 2: Find the slopes of the tangents

For the first curve $x=y^2$:

Differentiating with respect to $x$:

For the second curve $xy=k$:

Differentiating with respect to $x$ using the product rule:

Step 3: Apply the condition for orthogonality

At the point of intersection, $m_1\cdot m_2=-1$:

The $y$ terms cancel out:

Step 4: Solve for $k$

We know from Step 1 that $x=k^{2/3}$. Substituting $x=\frac{1}{2}$:

Cubing both sides:

Taking the square root:

Conclusion

The value of $k$ is $\frac{1}{2\sqrt{2}}$.

The correct option is (a).