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JEE Mains | PYQ

Let a circle of radius \( 4 \) pass through the origin \( O \), the point \( A(-\sqrt{3}\,a, 0) \) and \( B(0, -\sqrt{2}\,b) \), where \( a \) and \( b \) are real parameters and \( ab \neq 0 \). Then the locus of the centroid of \( \triangle OAB \) is a circle of radius:

Let the line \( L_1 : x + 3 = 0 \) intersect the lines \( L_2 : x - y = 0 \) and \( L_3 : 3x + y = 0 \) at the points \( A \) and \( B \), respectively. Let the bisector of the obtuse angle between the lines \( L_2 \) and \( L_3 \) intersect the line \( L_1 \) at the point \( C \). Then \( BC^2 : AC^2 \) is equal to:

Let \( A(1,2) \) and \( C(-3,-6) \) be two diagonally opposite vertices of a rhombus, whose sides \( AD \) and \( BC \) are parallel to the line \( 7x - y = 14 \). If \( B(\alpha,\beta) \) and \( D(\gamma,\delta) \) are the other two vertices, then \( |\alpha + \beta + \gamma + \delta| \) is equal to:

Let the mid points of the sides of a triangle \( ABC \) be \( \left(\dfrac{5}{2},7\right) \), \( \left(\dfrac{5}{2},3\right) \) and \( (4,5) \). If its incentre is \( (h,k) \), then \( 3h+k \) is equal to:

Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) on the line \( x + 2\sqrt{2}\,y = 4 \). If the co-ordinates of the vertex \( A \) are \( (\alpha, \beta) \) then the greatest integer less than or equal to \( \lvert \alpha + \sqrt{2}\beta \rvert \) is:

If a straight line drawn through the point of intersection of the lines \( 4x+3y-1=0 \) and \( 3x+4y-1=0 \), meets the co-ordinate axes at the points \( P \) and \( Q \), then the locus of the mid-point of \( PQ \) is:

A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to: