Maths-
General
Easy
Question
The area of the trapezium whose vertices lie on the parabola
and its diagonals pass through
and having length
units each is
The correct answer is: ![75 divided by 4](data:image/png;base64,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)
Related Questions to study
maths-
The Locus of centre of circle which touches given circle externally and the given line is
The Locus of centre of circle which touches given circle externally and the given line is
maths-General
maths-
If the length of tangents from
to the circles
and
are equal then ![10 a minus 2 b equals](data:image/png;base64,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)
If the length of tangents from
to the circles
and
are equal then ![10 a minus 2 b equals](data:image/png;base64,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)
maths-General
maths-
The line
cuts the circle
in
and Q Then the equation of the smaller circle passing through
and ![Q](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAOCAYAAAD0f5bSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAMyZLetQAAALtJREFUeNpjYMAO3IB4BxB/A+IfQHwYiB0Z8IAOIN4AxKZAzATFfkD8EoitsWnIA+KZOAwD2XQcXVAWiK9ATcYFPgAxG7JAKxDnMOAHu4HYAFkAZIs8AU0PgZgLxmGChhQ+wAJ1HorAJwKagoB4IbrgNwKBcBTdPwxQUxJxaKgE4i5sEhpA/BiI46HOBQE9IF4MDVmcwByI90OTDgj/B+IsBhJBOBA/B+IzQJwMxKLEauQB4nIgvgZNyAwAqaghgYyCZAkAAABJdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1pPlE8L21pPjwvbWF0aD715aU4AAAAAElFTkSuQmCC)
The line
cuts the circle
in
and Q Then the equation of the smaller circle passing through
and ![Q](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAOCAYAAAD0f5bSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAMyZLetQAAALtJREFUeNpjYMAO3IB4BxB/A+IfQHwYiB0Z8IAOIN4AxKZAzATFfkD8EoitsWnIA+KZOAwD2XQcXVAWiK9ATcYFPgAxG7JAKxDnMOAHu4HYAFkAZIs8AU0PgZgLxmGChhQ+wAJ1HorAJwKagoB4IbrgNwKBcBTdPwxQUxJxaKgE4i5sEhpA/BiI46HOBQE9IF4MDVmcwByI90OTDgj/B+IsBhJBOBA/B+IzQJwMxKLEauQB4nIgvgZNyAwAqaghgYyCZAkAAABJdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1pPlE8L21pPjwvbWF0aD715aU4AAAAAElFTkSuQmCC)
maths-General
maths-
If the angle between two equal circles with centres
,
is 12
then the radius of the circle is
If the angle between two equal circles with centres
,
is 12
then the radius of the circle is
maths-General
maths-
The equation of the circle passing through the origin and the points of intersection of the circles
![x to the power of 2 end exponent plus y to the power of 2 end exponent plus 4 x minus 2 y minus 4 equals 0](data:image/png;base64,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)
The equation of the circle passing through the origin and the points of intersection of the circles
![x to the power of 2 end exponent plus y to the power of 2 end exponent plus 4 x minus 2 y minus 4 equals 0](data:image/png;base64,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)
maths-General
maths-
The locus of a point such that difference of the squares of the tangents from it to two given circles is constant, is given by
The locus of a point such that difference of the squares of the tangents from it to two given circles is constant, is given by
maths-General
maths-
Radical centre of the circles
![x to the power of 2 end exponent plus y to the power of 2 end exponent plus 4 x minus 6 y plus 4 equals 0](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALUAAAATCAYAAADWFkxaAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAupJREFUeNrtWk1kXFEUviIiKkrEqOqiRERURTdVNapCRFRFlKqqqgpdjC5mX9VFlepiVEQYVVVVpaqLqGwqKiqiRERUVcmyi2y6qIoRZXqO+R7X633pfXPf/fF6Pz5m7pvrnffdc88957wRIj/a4D5xjTgi/CAUOyJKhB5ijbgZ7YgoG1rRjogy4TxxNdoRURYcRS47/B/ZcRF5fCg4TXxN/ImTagkb3Dds63SYuEj8BDYxZoRR4js4lE+4tGOA+CUgp74FJx6VFvoa8aNnu1zotEKcSW2iFdPIuNzlzijyQV3bwdFgLhCnPkHccHCfEHW6TJxXjD8mXk0PsiEPFT++j2sJ2JHGLIv0lHhFMV4nPnBoR4KqFAmy5urqVwSaqgXsUkfXOpniDXFKMT6Ba3+B22JHpO83iQuKB02zaGe6QXyUGjtE/EYccmgHo4+4TTyuMVdHvyLwVTPl0tHRh05Z66eznj9wL9X9d1UTpokNfJ40zVMMRLqAAkjGPeJdD8cqR9/bmnNt65eghVz6LXFPdF5AbSCnLlJHWzqZ4PcB1/azLrxHBb2luZtt7Ea+72fpe4W4Q+x3bMc4Oit5FjqPft3a1capwE7bKzovoKqIwDUDHV3qZMOpM99V1OHx454Ljz3pcwN2ubaD20UjOefa1C8Bt/COKcZPEb8XqKNNnbrdQLsZ6Ud/VvpRRbLd0CxEbDo1t6aGwR1EI9d25BXdtn5ysd6jeQSb6GhLJ9NCcVoxPqUqFPmhl1BIDCBHG/Lo1M9Fpxf5kng9gFbVv+a60C9BLaOrMYaoWZSObU8aH4RLxCeK8WfpQFLB7pcXgRvaLzw6Ux0tqW3PJ4bOXFf6yZU+/y1gDjk14wy0mixQxxCdmvEBz9ULLe6I1N8k+lBFq3K0Vxmh3gVmIcyMCAtthYP50K+CiPULxRMv6rmAdLTp1IPYqC3UDE2cjsGDF2FdREQdSwQ+zs9GGaKOZcFJ0fmjUkTUMTf+ANlLH5MmjLm0AAAA73RFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bXN1cD48bWk+eTwvbWk+PG1uPjI8L21uPjwvbXN1cD48bW8+KzwvbW8+PG1uPjQ8L21uPjxtaT54PC9taT48bW8+LTwvbW8+PG1uPjY8L21uPjxtaT55PC9taT48bW8+KzwvbW8+PG1uPjQ8L21uPjxtbz49PC9tbz48bW4+MDwvbW4+PC9tYXRoPgyLeAkAAAAASUVORK5CYII=)
Radical centre of the circles
![x to the power of 2 end exponent plus y to the power of 2 end exponent plus 4 x minus 6 y plus 4 equals 0](data:image/png;base64,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)
maths-General
maths-
The radical axis of circles
and
‐5x
is
The radical axis of circles
and
‐5x
is
maths-General
maths-
If the circle
bisects the circumference ofthe circle
then ![k to the power of 2 end exponent minus 1 equals](data:image/png;base64,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)
If the circle
bisects the circumference ofthe circle
then ![k to the power of 2 end exponent minus 1 equals](data:image/png;base64,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)
maths-General
Maths-
Two circles of equal radii
cut orthogonally If their centres are
and
, then ![r equals](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAKCAYAAABBq/VWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAAFpJREFUeNpjYECAeiAuB+IMID4DxD+gfKqCVUD8FIh7gFgJj7r/RGCc4BYQBzHQEDAB8TcGGgNzIN5PpFqygysSiOfT2ieTgDiZ1pasA2IvWlvyDog5GIYyAACwWx66pgU/iwAAAFN0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWk+cjwvbWk+PG1vPj08L21vPjwvbWF0aD7WCX2yAAAAAElFTkSuQmCC)
Two circles of equal radii
cut orthogonally If their centres are
and
, then ![r equals](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAKCAYAAABBq/VWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAAFpJREFUeNpjYECAeiAuB+IMID4DxD+gfKqCVUD8FIh7gFgJj7r/RGCc4BYQBzHQEDAB8TcGGgNzIN5PpFqygysSiOfT2ieTgDiZ1pasA2IvWlvyDog5GIYyAACwWx66pgU/iwAAAFN0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWk+cjwvbWk+PG1vPj08L21vPjwvbWF0aD7WCX2yAAAAAElFTkSuQmCC)
Maths-General
maths-
The number of points such that the tangents from it to three given circles are equal in length, is
The number of points such that the tangents from it to three given circles are equal in length, is
maths-General
maths-
The angle at which the circles
intersect is
The angle at which the circles
intersect is
maths-General
maths-
The eccentricity of the ellipse which meets the straight line
on the axis of X and the straight line
on the axis
and whose axis lie along the coordinate axes is
The eccentricity of the ellipse which meets the straight line
on the axis of X and the straight line
on the axis
and whose axis lie along the coordinate axes is
maths-General
physics-
A point source of length S is located on a wall. A plane mirror M having length l is moving parallel to the wall with constant velocity v. The bright patch formed on the wall by reflected light will
![](data:image/png;base64,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)
![](data:image/png;base64,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)
A point source of length S is located on a wall. A plane mirror M having length l is moving parallel to the wall with constant velocity v. The bright patch formed on the wall by reflected light will
![](data:image/png;base64,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)
![](data:image/png;base64,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)
physics-General
physics-
A uniform ring is rotating about vertical axis with angular velocity
initially. A point insect (S) having the same mass as that of the ring starts walking from the lowest point P1 and finally reaches the point P2 (as shown in figure). The final angular velocity of the ring will be equal to
![](data:image/png;base64,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)
A uniform ring is rotating about vertical axis with angular velocity
initially. A point insect (S) having the same mass as that of the ring starts walking from the lowest point P1 and finally reaches the point P2 (as shown in figure). The final angular velocity of the ring will be equal to
![](data:image/png;base64,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)
physics-General