If the tangent to ellipse at the point ( a cos alpha , b sin alpha) meets the coordinate axes at P and Q and M is midpoint of PQ, find the coordinates of M and hence prove that as alpha varies,the locus of M is the curve a 2 /x 2 + b 2 /y 2 =4 .

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Vidhi Shah · Answer

Tangent at p:(can`t be bothered typing out the derivation...)[xsec@ / a] + [ytan@ / b] = 1but when the tangent crosses the x axis, y = 0therefore x = a/sec@ = acos@therefore T has co-ordinates (acos@, 0)PS and PS` will be inclined at the same angle to PT if PS/PS` = ST/S`T (from question a).ST/S`T = (ae - acos@)/(acos@ - ae)= (e - cos@)/(cos@ - e)PS/PS` = ..... ( i used the distance formula here- i see no other way of doing it, uless there are similar triangles somewhere that I missed, but yeah... and i CANNOT be stuffed typing the distance formula out- so...)= (e - cos@)/(cos@ - e)Therefore PS:PS` = ST/S`T, and thus Angle SPT = angle S`PTtherefore the lines PS and PS` are inclined to the tangent at P at equal angles.

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If the tangent to ellipse at the point ( a cos alpha , b sin alpha) meets the coordinate axes at P and Q and M is midpoint of PQ, find the coordinates of M and hence prove that as alpha varies,the locus of M is the curve a 2 /x 2 + b 2 /y 2 =4 .

If the tangent to ellipse at the point ( a cos alpha , b sin alpha) meets the coordinate axes at P and Q and M is midpoint of PQ, find the coordinates of M and hence prove that as alpha varies,the locus of M is the curve a²/x²+ b²/y²=4 .

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If the tangent to ellipse at the point ( a cos alpha , b sin alpha) meets the coordinate axes at P and Q and M is midpoint of PQ, find the coordinates of M and hence prove that as alpha varies,the locus of M is the curve a 2 /x 2 + b 2 /y 2 =4 .

If the tangent to ellipse at the point ( a cos alpha , b sin alpha) meets the coordinate axes at P and Q and M is midpoint of PQ, find the coordinates of M and hence prove that as alpha varies,the locus of M is the curve a2/x2+ b2/y2=4 .

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If the tangent to ellipse at the point ( a cos alpha , b sin alpha) meets the coordinate axes at P and Q and M is midpoint of PQ, find the coordinates of M and hence prove that as alpha varies,the locus of M is the curve a²/x²+ b²/y²=4 .