Question

Find the equation of the ellipse whose focus is (1,0) the directrix is x+y+1=0 and eccentricity is 1/√2​

Answer 1

Answer:

3x²+3y²-2xy+6x-2y+3 =0

Step-by-step explanation:

directrix equation : x+y+1 =0

focus : (1 , 0)

eccenticity : 1/ √√ 2

Let P=(h,k)

(h+1)² +k² = 1/2(h+k+1)² /2

4(x²+y²+2x+1) = x²+y²+1+2xy+2y+2x

3x²+3y²-2xy+6x-2x+3 =0

Answer 2

Question:-

Find the equation of the ellipse whose focus is (1,0) the directrix is x+y+1=0 and eccentricity is 1/√2.

Given:-

• Focus is (1,0).

• Directix is x + y + 1 = 0.

• e is 1/√2.

Find:-

• The Equation of the ellipse.

Solution:-

Let, S be the focus of the ellipse and e be the eccentricity of the ellipse.

Consider that P(x,y) be any point on the ellipse, then by the definition of ellipse.

${ \boxed { \sf \large \red{SP = e × PM.}}}$

$\sf \large \sqrt{(x - 1) {}^{2} + (y - 0) { }^{2} } = \frac{1}{ \sqrt{2} } \bigg( \frac{x + y + 1}{ \sqrt{1 {}^{2} + 1 {}^{2} } } \bigg)$

$\sf \large\sqrt{(x - 1) {}^{2} + (y) {}^{2} } = \frac{1}{2} (x + y + 1)$

$\sf \large(x - 1) {}^{2} + (y) {}^{2} = \frac{1}{4} (x + y + 1) {}^{2}$

$\sf \large3x {}^{2} + 3y {}^{2} - 2xy - 10x - 2y + 3 = 0.$

Answer:-

$\sf \large \: \red{ The \: required \: equation \: of \: ellipse \: is \: }\\ \sf \large \: 3x {}^{2} + 3y {}^{2} - 2xy - 10x - 2y + 3 = 0.$

Hope you have satisfied. ⚘