The number of pairs (a, b) of positive real n | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The given inequalities are $a^4+b^4  1$ for positive real numbers $a$ and $b$.Consider the curves $x^2+y^2=1$ (a circle of radi

Vedclass · Accepted Answer

More than $2$

Vedclass · Answer

(D) The given inequalities are $a^4+b^4  1$ for positive real numbers $a$ and $b$.Consider the curves $x^2+y^2=1$ (a circle of radius $1$) and $x^4+y^4=1$ (a squircle) in the first quadrant where $x, y > 0$.For any point $(x, y)$ on the circle $x^2+y^2=1$,we have $x^4+y^4 = (x^2+y^2)^2 - 2x^2y^2 = 1 - 2x^2y^2$. Since $x, y > 0$,$x^2y^2 > 0$,which implies $x^4+y^4 This means the region $a^4+b^4  1$ lies outside the circle. However,the curve $a^4+b^4=1$ actually encloses a larger area than the circle $a^2+b^2=1$ in the first quadrant.Specifically,for $a, b > 0$,the region $a^2+b^2 > 1$ and $a^4+b^4 Since this region has a non-zero area,there are infinitely many pairs $(a, b)$ of positive real numbers satisfying these inequalities.Thus,the number of such pairs is more than $2$.

The number of pairs $(a, b)$ of positive real numbers satisfying $a^4+b^4 < 1$ and $a^2+b^2 > 1$ is

Similar Questions

If $A(-a, 0)$ and $B(a, 0)$ are two fixed points,then the locus of the point $P(x, y)$ on which the line segment $AB$ subtends a right angle is:

If a circle of a constant radius $6$ passes through the origin $O$ and meets the coordinate axes at $A$ and $B$, then find the locus of the centroid of $\triangle OAB$.

The locus of the centre of the circle which cuts a chord of length $2a$ from the positive $x$-axis and passes through a point on the positive $y$-axis at a distance $b$ from the origin is:

$A$ circle touches the $x$-axis and cuts off a chord of length $2l$ from the $y$-axis. The locus of the centre of the circle is

The locus of a point which is at a distance of $4$ units from $(3, -2)$ in the $xy$-plane is

Vedclass Test Series

Exam Paper Generator

Online Exam Module