find the locus of intersection of tangents to the circle x=acosQ ,Y=ASINQ at the pints whose parametric angles differ be - Maths - Conic Sections

Question

find the locus of intersection of tangents to the circle x=acosQ
,Y=ASINQ at the pints whose parametric angles differ be pie/3 and
pie/2

Lovina Kansal · Accepted Answer

Dear student
Let one of the points on the circle be A(acosQ,asinQ)Then the other point will be BacosQ+π3,asinQ+π3∴Equation of tangent at A=xcosQ+ysinQ=a     
(1)Equation of tangent at B=xcosQ+π3+ysinQ+π3=a     
(2)From eq(2)⇒x12cosQ-32sinQ+y12sinQ+32cosQ=a⇒12xcosQ+ysinQ-32xsinQ-ycosQ=a⇒xsinQ-ycosQ=-a3      
(3)∴Square and add (1) and (3)xcosQ+ysinQ2+xsinQ-ycosQ2=a2+a23⇒3x2+3y2=4a2The locus of the point of intersection of the tangents 3x2+3y2-4a2=0
Regards

find the locus of intersection of tangents to the circle x=acosQ ,Y=ASINQ at the pints whose parametric angles differ be pie/3 and pie/2