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Question

If $alpha comma beta$ are the eccentric angles of the extremities of a focal chord of the ellipse $fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 9 end fraction equals 1$ , then tan $left parenthesis alpha divided by 2 right parenthesis t a n invisible function application left parenthesis beta divided by 2 right parenthesis equals$

$\frac{\sqrt{7} + 4}{\sqrt{7} - 4}$
$- \frac{9}{23}$
$\frac{\sqrt{5} - 4}{\sqrt{5} + 4}$
$\frac{8 \sqrt{7} - 23}{9}$

The correct answer is: $fraction numerator 8 square root of 7 minus 23 over denominator 9 end fraction$

The equation of the ellipse is of the form $x to the power of 2 end exponent divided by a to the power of 2 end exponent plus y to the power of 2 end exponent divided by b to the power of 2 end exponent equals 1 text where end text a to the power of 2 end exponent equals 16 comma b to the power of 2 end exponent equals 9$
$therefore$ the eccentricity e = $square root of 1 minus fraction numerator 9 over denominator 16 end fraction end root$ = $blank fraction numerator square root of 7 over denominator 4 end fraction$ .
Let P(4 cos $alpha$ , 3 sin $alpha$ ) and Q (4 cos $beta$ , 3 sin $beta$ ) be a focal chord of the ellipse passing through the focus at ( $square root of 7$ , 0).
Then $fraction numerator 3 sin invisible function application beta over denominator 4 cos invisible function application beta minus square root of 7 end fraction$ = $fraction numerator 3 sin invisible function application alpha over denominator 4 cos invisible function application alpha minus square root of 7 end fraction$
$rightwards double arrow$  $fraction numerator sin invisible function application left parenthesis alpha minus beta right parenthesis over denominator sin invisible function application alpha minus sin invisible function application beta end fraction$ = $fraction numerator square root of 7 over denominator 4 end fraction$
$rightwards double arrow$  $fraction numerator cos invisible function application left square bracket left parenthesis alpha minus beta right parenthesis divided by 2 right square bracket over denominator cos invisible function application left square bracket left parenthesis alpha plus beta right parenthesis divided by 2 right square bracket end fraction$ = $fraction numerator square root of 7 over denominator 4 end fraction$
$rightwards double arrow$ tan $open parentheses fraction numerator alpha over denominator 2 end fraction close parentheses$ tan $open parentheses fraction numerator beta over denominator 2 end fraction close parentheses$ = $fraction numerator square root of 7 minus 4 over denominator square root of 7 plus 4 end fraction$ = $fraction numerator 23 minus 8 square root of 7 over denominator negative 9 end fraction$ .

Related Questions to study

If P( $theta$ ), Q $open parentheses theta plus fraction numerator pi over denominator 2 end fraction close parentheses$ are points on ellipse and $alpha$ is angle between normals at P and Q then -

If P( $theta$ ), Q $open parentheses theta plus fraction numerator pi over denominator 2 end fraction close parentheses$ are points on ellipse and $alpha$ is angle between normals at P and Q then -

The Maxwell's four equations are written as:
i) $not stretchy contour integral stack E with rightwards arrow on top. stack d s with rightwards arrow on top equals fraction numerator q subscript 0 end subscript over denominator epsilon subscript 0 end subscript end fraction$
ii) $not stretchy contour integral stack B with rightwards arrow on top. stack d s with rightwards arrow on top equals 0$
iii) $not stretchy contour integral stack E with rightwards arrow on top. stack d l with rightwards arrow on top equals fraction numerator d over denominator d t end fraction not stretchy contour integral stack B with rightwards arrow on top. stack d s with rightwards arrow on top$
iv) $not stretchy contour integral stack B with rightwards arrow on top. stack d s with rightwards arrow on top equals mu subscript 0 end subscript epsilon subscript 0 end subscript fraction numerator d over denominator d t end fraction not stretchy contour integral stack E with rightwards arrow on top. stack d s with rightwards arrow on top$
The equations which have sources of $stack E with rightwards arrow on top$ and $stack B with rightwards arrow on top$

The Maxwell's four equations are written as:
i) $not stretchy contour integral stack E with rightwards arrow on top. stack d s with rightwards arrow on top equals fraction numerator q subscript 0 end subscript over denominator epsilon subscript 0 end subscript end fraction$
ii) $not stretchy contour integral stack B with rightwards arrow on top. stack d s with rightwards arrow on top equals 0$
iii) $not stretchy contour integral stack E with rightwards arrow on top. stack d l with rightwards arrow on top equals fraction numerator d over denominator d t end fraction not stretchy contour integral stack B with rightwards arrow on top. stack d s with rightwards arrow on top$
iv) $not stretchy contour integral stack B with rightwards arrow on top. stack d s with rightwards arrow on top equals mu subscript 0 end subscript epsilon subscript 0 end subscript fraction numerator d over denominator d t end fraction not stretchy contour integral stack E with rightwards arrow on top. stack d s with rightwards arrow on top$
The equations which have sources of $stack E with rightwards arrow on top$ and $stack B with rightwards arrow on top$

physics-General

The line x cos $alpha$ + y sin $alpha$ = p touches the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , if :

The line x cos $alpha$ + y sin $alpha$ = p touches the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , if :

parallel

If $alpha$ and $beta$ are the eccentric angles of extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is -

If $alpha$ and $beta$ are the eccentric angles of extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is -

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If x cos $alpha$ + y sin $alpha$ = p is a tangent to the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , then -

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parallel

In Young's double slit experiment, 12 fringes are obtained to be formed in a certain segment of the screen when light of wavelength 600 nm is used. If wavelength of light is changed to 400 nm, number of fringes observed in the same segment of the screen is given by

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physics-General

Intensity of central bright fringe due to interference of two identical coherent monochromatic sources is I. If one of the source is switched off, then intensity of central bright fringe becomes

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physics-General

If P (a cos $theta$ , bsin $theta$ ) is a point on an ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , then ' $theta$ ' is –

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parallel

If the normal at the point P( $theta$ ) to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 14 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 5 end fraction equals 1$ intersects it again at the point Q(2 $theta$ ) then cos $theta$ =

If the normal at the point P( $theta$ ) to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 14 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 5 end fraction equals 1$ intersects it again at the point Q(2 $theta$ ) then cos $theta$ =

If a tangent to ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction blank$ + $fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction$ = 1 makes an angle $alpha$ with x- axis, then square of length of intercept of tangent cut between axes is-

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Vibrating tuning fork of frequency $n$ is placed near the open end of a long cylindrical tube. The tube has a side opening and is fitted with a movable reflecting piston. As the piston is moved through $8.75 blank c m comma$ the intensity of sound changes from a maximum to minimum. If the speed of sound is $350 blank m divided by s.$ then $n$ is

Vibrating tuning fork of frequency $n$ is placed near the open end of a long cylindrical tube. The tube has a side opening and is fitted with a movable reflecting piston. As the piston is moved through $8.75 blank c m comma$ the intensity of sound changes from a maximum to minimum. If the speed of sound is $350 blank m divided by s.$ then $n$ is

physics-General

parallel

PQ and QR are two focal chords of an ellipse and the eccentric angles of P,Q,R and
$2 alpha$ , $2 beta comma blank 2 gamma$ respectively then tan $beta$ tan $gamma$ is equal to -

PQ and QR are two focal chords of an ellipse and the eccentric angles of P,Q,R and
$2 alpha$ , $2 beta comma blank 2 gamma$ respectively then tan $beta$ tan $gamma$ is equal to -

The radius of the circle passing through the points of intersection of ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction$ = 1 and x²– y² = 0 is -

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If $alpha comma beta$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is -

If $alpha comma beta$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is -

parallel

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