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Maths-

General

Easy

Question

If the chord joining two points whose eccentric angles are $alpha$ and $beta$ cuts the major axis of an ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction$ + $fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction$ = 1 at a distance c from the centre, then tan $alpha divided by 2$ . tan $beta divided by 2$ is equal to

1
c
$\frac{c + a}{c - a}$
$\frac{c - a}{c + a}$

The correct answer is: $fraction numerator c plus a over denominator c minus a end fraction$

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P(a cos $theta$ , b sin $theta$ ) is any point on the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction$ + $fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction$ = 1 $left parenthesis a greater than b right parenthesis$ Now if this ellipse is rotated anticlockwise by 90º and P reaches $straight P to the power of straight prime$ then co-ordinate of $straight P to the power of straight prime$ are -

P(a cos $theta$ , b sin $theta$ ) is any point on the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction$ + $fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction$ = 1 $left parenthesis a greater than b right parenthesis$ Now if this ellipse is rotated anticlockwise by 90º and P reaches $straight P to the power of straight prime$ then co-ordinate of $straight P to the power of straight prime$ are -

Normal to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 84 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 49 end fraction equals 1$ intersects the major and minor axis at P and Q respectively then locus of the point dividing segment PQ in 2 : 1 is -

Normal to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 84 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 49 end fraction equals 1$ intersects the major and minor axis at P and Q respectively then locus of the point dividing segment PQ in 2 : 1 is -

Given graph gives variation of $fraction numerator P V over denominator T end fraction$ with P for 1gm of oxygen at two different temperatures T₁ and T₂. If density of oxygen $1.427 k g divided by m to the power of 3 end exponent.$ The value of $fraction numerator P V over denominator T end fraction$ at point A and relation $b divided by w T subscript 1 end subscript & T subscript 2 end subscript text is end text$

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