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An isosceles ?ABC is inscribed in the circle x2 + y2 = 16. From points A, B and C three ordinates are drawn which cut the ellipse x2/16+y2/9=1 at the points P, Q and R respectively such that A and P are on the opposite sides, Q and B are on the same side and C and R are on the same side of the major axis. If ?PQR is right angled at P and ? is the smallest angle of the ?ABC, then find the value of 16tan2? + 8 tan ? + 9.

Asked by savithakruthik9945 | 15 Jul, 2022, 21:34: PM

Expert Answer

SInce comma space straight A comma space straight B comma space straight C space lie space on space straight a space circle space straight x squared plus straight y squared equals 16 Let space straight A equals left parenthesis 4 space cosθ comma space 4 space sinθ right parenthesis comma space straight B equals left parenthesis 4 space cos left parenthesis straight theta plus straight alpha right parenthesis comma space 4 space sin left parenthesis straight theta plus straight alpha right parenthesis right parenthesis comma space straight C equals left parenthesis 4 space cos left parenthesis straight theta minus straight alpha right parenthesis comma space 4 space sin left parenthesis straight theta minus straight alpha right parenthesis right parenthesis rightwards double arrow straight P equals left parenthesis 4 space cosθ comma space minus 3 space sinθ right parenthesis comma space straight B equals left parenthesis 4 space cos left parenthesis straight theta plus straight alpha right parenthesis comma space 3 space sin left parenthesis straight theta plus straight alpha right parenthesis right parenthesis comma space straight C equals left parenthesis 4 space cos left parenthesis straight theta minus straight alpha right parenthesis comma space 3 space sin left parenthesis straight theta minus straight alpha right parenthesis right parenthesis As space PQR thin space is space right space angled space triangle space at space straight P rightwards double arrow Slope space of space PQ space cross times space Slope space of space PR space equals space minus 1 rightwards double arrow 9 over 16 open square brackets fraction numerator sin space straight theta space plus space sin left parenthesis straight theta plus straight alpha right parenthesis over denominator cos space straight theta space minus space cos left parenthesis straight theta plus straight alpha right parenthesis end fraction close square brackets open square brackets fraction numerator sin space straight theta space plus space sin left parenthesis straight theta minus straight alpha right parenthesis over denominator cos space straight theta space minus space cos left parenthesis straight theta minus straight alpha right parenthesis end fraction close square brackets equals negative 1 rightwards double arrow 9 over 16 open square brackets fraction numerator sin open parentheses straight theta plus begin display style straight alpha over 2 end style close parentheses space cos open parentheses begin display style straight alpha over 2 end style close parentheses over denominator sin open parentheses begin display style straight alpha over 2 end style close parentheses space sin open parentheses straight theta plus begin display style straight alpha over 2 end style close parentheses end fraction cross times fraction numerator sin open parentheses straight theta minus begin display style straight alpha over 2 end style close parentheses space cos open parentheses begin display style straight alpha over 2 end style close parentheses over denominator sin open parentheses begin display style negative straight alpha over 2 end style close parentheses space sin open parentheses straight theta plus negative close parentheses end fraction close square brackets equals negative 1 rightwards double arrow 9 over 16 open parentheses negative cot squared straight alpha over 2 close parentheses equals negative 1 rightwards double arrow cot squared straight alpha over 2 equals 16 over 9 rightwards double arrow tan straight alpha over 2 equals 3 over 4 Angles space of space triangle ABC thin space are space straight alpha over 2 comma space straight alpha over 2 comma space straight pi minus straight alpha So comma space the space angles space are space tan to the power of negative 1 end exponent 3 over 4 comma space tan to the power of negative 1 end exponent 3 over 4 comma space straight pi minus 2 tan to the power of negative 1 end exponent 3 over 4 Therefore comma space tanθ equals 3 over 4 rightwards double arrow 16 space tan squared straight theta space plus space 8 space tanθ space plus 9 space equals 24

Answered by Renu Varma | 21 Jul, 2022, 11:22: AM

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