## Question

If α – β = constant, then the locus of the point of intersection of tangents at to the ellipse

### Solution

An ellipse

Let *R*(*h*, *k*) be the point of intersection of tangents at *P *and *Q*. Then,

Hence, the locus of *R*(*h*, *k*) is

Clearly, it represents an ellipse.

#### SIMILAR QUESTIONS

If C is the centre and A, B are two points on the conic

4*x*^{2} + 9*y*^{2} – 8*x* – 36*y* + 4 = 0 such that ∠*ACB* = π/2 then *CA*^{–2} +*CB*^{–2} is equal to

Ellipses which are drawn with the same two perpendicular lines as axes and with the sum of the reciprocals of squares of the lengths of their semi-major axis and semi-minor axis equal to a constant have only.

The eccentricity of the ellipse with centre at the origin which meets the straight line on the axis of *x* and the straight line on the axis of *y* and whose axes lie along the axes of coordinates is

The radius of the circle passing through the foci of the ellipse

9*x*^{2} + 16*y*^{2} = 144 and having its centre at (0, 3), is

An ellipse has OB as a semi-minor axis, *F*, *F*’ as its foci and the angle ∠*FBF*’ is a right angle. Then, the eccentricity of the ellipse is

The focus of an ellipse is (–1, –1) and the corresponding directix is *x* – *y* + 3 = 0. If the eccentricity of the ellipse is 1/2, then the coordinates of the centre of the ellipse are

The equation of the ellipse with its centre at (1, 2), one focus at (6, 2) and passing through (4, 6) is

Tangents are drawn to the ellipse and the circle *x*^{2} + *y*^{2} = *a*^{2} at the points where a common ordinate cuts them (on the same side of the *x*-axis). Then, the greatest acute angle between these tangents is given by

The area of the quadrilateral formed by the tangents at the end-points of latusrecta to the ellipse

Let *S*(3, 4) and *S*’(9, 12) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent to the ellipse is (1, –4), then the eccentricity of the ellipse is