## Question

The tangent at a point meets the auxiliany circle in two points. The chord joining them subtends a right angle at the centre. Then, the eccentricity of the ellipse is given by

### Solution

The equation of the tangent at

The combined equation of the chords joining the points of intersection of (i) and the auxiliary circle *x*^{2} + *y*^{2} = *a*^{2} is

The lines given by this equation are at right angles.

∴ Coeff. of *x*^{2} + Coeff. of *y*^{2} = 0

#### SIMILAR QUESTIONS

The number of values of *c* such that the straight line *y* = 4*x* + *c*touches the curve , is

, then *PF*_{1} + *PF*_{2} equals

An ellipse slides between two perpendicular straight lines. Then, the locus of its centre is *a/an*

The sum of the squares of the perpendicular on any tangent to the ellipse from two points on the minor axis, each at a distance from the centre is

The eccentric angle of a point on the ellipse whose distance from the centre of the ellipse is 2, is

If any tangent to the ellipse intercepts equal length *l*on the axes, then *l *=

The ellipse *x*^{2} + 4*y*^{2} = 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then, the equation of the ellipse is

A focus of an ellipse is at the origin. The directrix is the line *x* = 4 and the eccentricity is 1/2. Then the length of the semi-major axis, is

In an ellipse, the distance between its foci is 6 and minor axis is 8. The eccentricity is

If *F*_{1} and *F*_{2} be the feet of the perpendicular from the foci *S*_{1}and *S*_{2} of an ellipse on the tangent at any point *P* on the ellipse, then (*S*_{1}*F*_{1})(*S*_{2}*F*_{2}) is equal to