0

Guest

Courses New

The points of intersection of the circle x 2 + y 2 =a 2 with the parabolas y 2 =4ax and y 2 =-4ax form a rectangle whose area is ? The point of intersection of the tangents to the parabola y 2 =4x at the points where the circle (x-3) 2 +y 2 =9 meets the parabola ,other than the origin is ?

  1. The points of intersection of the circle x2+y2=a2 with the parabolas y2=4ax and y2=-4ax form a rectangle whose area is ?
  2. The point of intersection of the tangents to the parabola y2=4x at the points where the circle (x-3)2+y2=9 meets the parabola ,other than the origin is ?

Grade:11

1 Answers

Divyarth
11 Points
6 years ago
x² + y² = a² ----> y² = a² − x² 

Now use this value of y in each of the 2 equations of parabolas to find points of intersection. Since we have both parabolas y² = 4ax and y² = −4ax, we can assume a > 0 

y² = 4ax 
a² − x² = 4ax 
x² + 4ax = a² 
x² + 4ax + 4a² = a² + 4a² 
(x + 2a)² = 5a² 
x + 2a = ± a√5 
x = (−2±√5)a 

Any point on circle has x-coordinate between −a and a 
x = (√5−2)a 

y² = a² − x² = a² − (9−4√5)a² = 4(√5−2)a² 
y = ± 2√(√5−2)a 

-------------------------------- 

y² = −4ax 
a² − x² = −4ax 
x² − 4ax = a² 
x² − 4ax + 4a² = a² + 4a² 
(x − 2a)² = 5a² 
x − 2a = ± a√5 
x = (2±√5)a 

Any point on circle has x-coordinate between −a and a 
x = (2−√5)a 

y² = a² − x² = a² − (9−4√5)a² = 4(√5−2)a² 
y = ± 2√(√5−2)a 

-------------------------------- 

Width of rectangle = difference between x-coordinates: 
(√5−2)a − (2−√5)a = 2(√5−2)a 

Height of rectangle = difference between y-coordinates: 
2√(√5−2)a − (−2√(√5−2)a) = 4√(√5−2)a 

Area of rectangle 
2(√5−2)a * 4√(√5−2)a 
= (2*4) (√5−2)√(√5−2) * a² 
= 8 (√5−2)^(3/2) a²

Think You Can Provide A Better Answer ?

Other Related Questions on Analytical Geometry

View all Questions »

ASK QUESTION

Get your questions answered by the expert for free

Answer and earn gift vouchers

Win Gift vouchers upto Rs 500/-

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Click Here Know More

Complete Self Study Package designed by Industry Leading Experts

Click Here Know More

Live 1-1 coding classes to unleash the Creator in your Child

Click Here Know More

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Click Here Know More