the ellipse with its focusand centre and passing through point given how to find equation of ellipse
jadhav ram , 5 Years ago
Grade 12
Analytical Geometry> the ellipse with its focus and centre and...
1 Answers
Aditya Gupta
Last Activity: 5 Years ago
hello jadhav. this is an excellent ques tbh. lets say one of the given focus is (r, s) and centre is (c, d). then we can find the coordinates of the second focus by using the fact that the centre of an ellipse lies in the centre (or mid point) of both the foci. so assume that coordinates of the second focus are (p, q). then by section formula, we can write
(r+p)/2= c and (s+q)/2= d..........from these we obtain p and q.
now, by the definition of ellipse, for every point (x, y) the sum of its distances from two other points (the foci) is constant.
so that √[(x – r)^2+(y – s)^2] + √[(x – p)^2+(y – q)^2]= K (a const)
now, we are also given a point (say m, n) on the ellipse.. so substitute that value in above eqn.
we get √[(m – r)^2+(n – s)^2] + √[(m– p)^2+(n – q)^2]= K so that we obtain the value of K. so the eqn becomes
√[(x – r)^2+(y – s)^2] + √[(x – p)^2+(y – q)^2]= √[(m – r)^2+(n – s)^2] + √[(m– p)^2+(n – q)^2]
now if you wish you can be simplify this eqn by getting rid of the square roots.
kindly approve ;)
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