**Question 1:**

If the points of the intersection of the ellipses and are the end points of the conjugate diameters of the former, prove that :

**Solution 1:**

The locus of the middle points of a system of parallel chords of an ellipse is a line passing through the centre of the ellipse. This is called the diameter of the ellipse and two diameters of the ellipse are said to be conjugate if each bisects the chords, parallel to the other. The condition for this is that the product of their slopes should be equal to .

Now, equation of the lines joining the centre to the points of intersection of the given ellipses is

…call this equation I;

If , are the slopes of the lines represented by Equation I, then

Since I represents a pair of conjugate diameters,

Thus,

**Question 2:**

Find the locus of the mid-points of the chords of the circle which are tangents to the hyperbola, .

**Solution 2:**

Let be the middle point of a chord of the circle .

Then, its equation is , that is, ….call this equation I.

Let I touch the hyperbola:

That is, …call this equation II.

at the point say, then I is identical with

….call this equation III.

Thus,

Since lies on the hyperbola II,

.

Hence, the required locus of is .

**Question 3:**

If P be a point on the ellipse whose ordinate is , prove that the angle between the tangent at P and the focal chord through P is .

**Solution 3:**

Let the coordinates of P be so that . Equation of the tangent at P is .

Slope of the tangent is equal to

Slope of the focal chord SP is .

If is the required angle, then

which in turn equals ,

More later,

I hope you like it…my students should be inspired to try Math on their own…initially, it is slow, gradual, painstaking, but the initial “roots” pay very very “rich dividends” later…

-Nalin Pithwa.

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