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1

### JEE Main 2021 (Online) 27th August Morning Shift

Numerical
Let the equation x2 + y2 + px + (1 $$-$$ p)y + 5 = 0 represent circles of varying radius r $$\in$$ (0, 5]. Then the number of elements in the set S = {q : q = p2 and q is an integer} is __________.

## Explanation

$$r = \sqrt {{{{p^2}} \over 4} + {{{{(1 - p)}^2}} \over 4} - 5} = {{\sqrt {2{p^2} - 2p - 19} } \over 2}$$

Since, $$r \in (0,5]$$

So, $$0 < 2{p^2} - 2p - 19 \le 100$$

$$\Rightarrow p \in \left[ {{{1 - \sqrt {239} } \over 2},{{1 - \sqrt {39} } \over 2}} \right) \cup \left( {{{1 + \sqrt {39} } \over 2},{{1 + \sqrt {239} } \over 2}} \right]$$

so, number of integral values of p2 is 61.
2

### JEE Main 2021 (Online) 26th August Morning Shift

Numerical
The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter d. Then d2 is equal to _____________.

## Explanation

Let point P(x, y)

A(0, 0), B(1, 0), C(0, 1), D(1, 1)

(PA)2 + (PB)2 + (PC)2 + (PD)2 = 18

$${x^2} + {y^2} + {x^2} + {(y - 1)^2} + {(x - 1)^2} + {y^2} + {(x - 1)^2} + {(y - 1)^2}$$ = 18

$$\Rightarrow 4({x^2} + {y^2}) - 4y - 4x = 14$$

$$\Rightarrow {x^2} + {y^2} - x - y - {7 \over 2} = 0$$

$$d = 2\sqrt {{1 \over 4} + {1 \over 4} + {7 \over 2}}$$

$$\Rightarrow {d^2} = 16$$
3

### JEE Main 2021 (Online) 17th March Morning Shift

Numerical
The minimum distance between any two points P1 and P2 while considering point P1 on one circle and point P2 on the other circle for the given circles' equations

x2 + y2 $$-$$ 10x $$-$$ 10y + 41 = 0

x2 + y2 $$-$$ 24x $$-$$ 10y + 160 = 0 is ___________.

## Explanation

$${S_1}:{(x - 5)^2} + {(y - 5)^2} = 9$$

Centre (5, 5), r1 = 3

$${S_2}:{(x - 12)^2} + {(y - 5)^2} = 9$$

Centre (12, 5), r2 = 3 So (P1P2)min = 1
4

### JEE Main 2021 (Online) 24th February Evening Slot

Numerical
If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle (x $$-$$ 2)2 + (y $$-$$ 3)2 = 25 at the point (5, 7) is A, then 24A is equal to _________.

## Explanation

This question is bonus if we consider poistive x axis.If we consider only x axis for this question then it is right question.

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