Let an ellipse E : x 2 a 2 + y 2 b 2 = 1 , a 2 > b 2 , passes through 3 2 , 1 and has eccentricity 1 3 . If a circle, centered at focus F ( α , 0 ) , α > 0 , of E and radius 2 3 , intersects E at two points P and Q , then

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Let an ellipse E : x 2 a 2 + y 2 b 2 = 1 , a 2 &#62; b 2 , passes through 3 2 , 1 and has eccentricity 1 3 . If a circle, centered at focus F ( &#945; , 0 ) , &#945; &#62; 0 , of E and radius 2 3 , intersects E at two points P and Q , then P Q 2 is equal to :

Quantrex Academy · Accepted Answer

Correct answer: C. 16 3

The ellipse x 2 a 2 + y 2 b 2 = 1 passes through the point 3 2 , 1 So, 3 2 a 2 + 1 b 2 = 1 &#160; &#160; &#160; . . . 1 and 1 - b 2 a 2 = 1 3 &#160; &#160; &#160; . . . 2 On solving equations 1 and 2 , we get &#8658; a 2 = 3 &#160; &#38; &#160; b 2 = 2 &#8658; x 2 3 + y 2 2 = 1 &#160; &#160; &#160; . . . 3 We know that, focii of the ellipse x 2 a 2 + y 2 b 2 = 1 is &#177; a e , &#160; 0 Here, a = 3 &#160; &#38; &#160; e = 1 3 &#8756; Its focus is 1 , 0 &#160; &#8757; &#945; &#62; 0 Now, equation of circle is ( x -

Let an ellipse E : x 2 a 2 + y 2 b 2 = 1 , a 2 > b 2 , passes through 3 2 , 1 and has eccentricity 1 3 . If a circle, centered at focus F ( α , 0 ) , α > 0 , of E and radius 2 3 , intersects E at two points P and Q , then P Q 2 is equal to :

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More from Ellipse

Let an ellipse E : x 2 a 2 + y 2 b 2 = 1 , a 2 &#62; b 2 , passes through 3 2 , 1 and has eccentricity 1 3 . If a circle, centered at focus F ( &#945; , 0 ) , &#945; &#62; 0 , of E and radius 2 3 , intersects E at two points P and Q , then P Q 2 is equal to :

Options

Correct answer

Step-by-step solution

More from Ellipse

Let an ellipse E : x 2 a 2 + y 2 b 2 = 1 , a 2 > b 2 , passes through 3 2 , 1 and has eccentricity 1 3 . If a circle, centered at focus F ( α , 0 ) , α > 0 , of E and radius 2 3 , intersects E at two points P and Q , then P Q 2 is equal to :