A Tangent Sloping at 30°
Difficulty: Medium · Topics: Geometry, Conic Sections, Ellipse
30° qiyalikdagi urinma
The ellipse dfracx^2a^2+dfracy^2b^2=1 has foci F(4, 0) and F'(-4, 0), and a circle is drawn with the segment F'F as a diameter. Let P be the intersection point of the ellipse and the circle that lies in the second quadrant.
graph "x":[-9.4,7],"y":[-4.6,5.4],"xTicks":[],"yTicks":[],"label":"An ellipse, the circle on F′F as diameter, and the tangent to the circle at P making 30° with the x-axis","items":["type":"ellipse","at":[0,0],"rx":5.464,"ry":3.722,"type":"circle","at":[0,0],"r":4,"type":"line","from":[-8,0],"to":[2.2,5.89],"ink":true,"type":"angle","at":[-8,0],"from":[-4,0],"to":[2.2,5.89],"label":"30°","r":1.25,"type":"point","at":[-2,3.464],"label":"P","anchor":"nw","type":"point","at":[4,0],"label":"F","anchor":"se","ink":true,"type":"point","at":[-4,0],"label":"F′","anchor":"se","ink":true]
If the tangent to the circle at P makes an angle of 30^° with the positive direction of the x-axis, what is the value of a^2+b^2? (Here a and b are constants with 0<√2 b<a.)
dfracx^2a^2+dfracy^2b^2=1 ellipsning fokuslari F(4, 0) va F'(-4, 0), F'F kesmani diametr qilib aylana chizilgan. P — ellips va aylananing ikkinchi chorakdagi kesishish nuqtasi.
graph "x":[-9.4,7],"y":[-4.6,5.4],"xTicks":[],"yTicks":[],"label":"An ellipse, the circle on F′F as diameter, and the tangent to the circle at P making 30° with the x-axis","items":["type":"ellipse","at":[0,0],"rx":5.464,"ry":3.722,"type":"circle","at":[0,0],"r":4,"type":"line","from":[-8,0],"to":[2.2,5.89],"ink":true,"type":"angle","at":[-8,0],"from":[-4,0],"to":[2.2,5.89],"label":"30°","r":1.25,"type":"point","at":[-2,3.464],"label":"P","anchor":"nw","type":"point","at":[4,0],"label":"F","anchor":"se","ink":true,"type":"point","at":[-4,0],"label":"F′","anchor":"se","ink":true]
Agar aylanaga P nuqtada o‘tkazilgan urinma x o‘qining musbat yo‘nalishi bilan 30^° burchak hosil qilsa, a^2+b^2 ning qiymati qancha? (Bu yerda a va b — 0<√2 b<a bo‘lgan o‘zgarmaslar.)