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Question

# Let the two foci of an ellipse be (−1,0) and (3,4) and the foot of perpendicular from the focus (3,4) upon a tangent to the ellipse be (4,6)The length of the semi-minor axis of the ellipse is

A
1
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B
22
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C
17
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D
43
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Solution

## The correct option is D 4√3Refer to the figure.Slope of line SP is given by,m1=y2−y1x2−x1∴m1=6−44−3∴m1=21=2Now, tangent through point P is perpendicular to line SP.Thus, slope of tangent is −12Thus, equation of tangent will be given by,y−y1=m(x−x1)∴y−6=−12(x−4)∴2(y−6)=−(x−4)∴2y−12=−x+4∴x+2y−16=0∴x+2y=16Dividing both sides by 16, we get,x16+2y16=1616∴x16+y8=1 (1)We know that equation of tangent to the ellipse through any point on the ellipse is given by,xx1a2+yy1b2=1∴4xa2+6yb2=1 (2)Comparing coefficients of x and y in equations (1) and (2), ∴4a2=116∴a2=64∴a=8Similarly, 6b2=18∴b2=48∴b=√48∴b=4√3Thus, length of the semi-minor axis of ellipse is 4√3

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