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Question

what is the sum of abscissa and ordinate of a point P where all variable chords of the parabola y^2=8x subtend right angle at the origin , are concurrent .

Answer 1

To find the sum of the abscissa and ordinate of a point \( P \) where all variable chords of the parabola \( y^2 = 8x \) that subtend a right angle at the origin are concurrent, follow these steps:

### Step 1: Equation of the Chord

A parabola \( y^2 = 8x \) can be parameterized using the standard form of its parametric equations:

\[

x = 2t^2, \quad y = 4t

\]

Let the parametric equations for two points on the parabola be \( (2t_1^2, 4t_1) \) and \( (2t_2^2, 4t_2) \). The equation of the chord joining these two points is:

\[

T = S_1

\]

where \( T \) is the equation of the chord and \( S_1 \) is the equation of the parabola evaluated at the point \( (x_1, y_1) \) on the parabola. For our parabola \( y^2 = 8x \), the chord equation becomes:

\[

y (y_1 - 4t_1) = 4(x - 2t_1^2)

\]

This simplifies to:

\[

y (4t_2 - 4t_1) = 4x - 8t_1^2

\]

or:

\[

y (t_2 - t_1) = x - 2(t_1^2 + t_1t_2 + t_2^2)

\]

### Step 2: Condition for Right Angle Chord

The condition for a chord of the parabola subtending a right angle at the origin is:

\[

t_1 t_2 = -1

\]

Using this condition in the equation of the chord, we get:

\[

y (t_2 - t_1) = x - 2(t_1^2 + t_2^2)

\]

Substitute \( t_2 = -\frac{1}{t_1} \):

\[

y \left(-\frac{1}{t_1} - t_1\right) = x - 2 \left(t_1^2 + \frac{1}{t_1^2}\right)

\]

Simplify:

\[

y \left(-\frac{1 + t_1^2}{t_1}\right) = x - 2 \left(t_1^2 + \frac{1}{t_1^2}\right)

\]

Rewriting:

\[

y \left(t_1^2 + 1\right) = -t_1 \left(x - 2 \left(t_1^2 + \frac{1}{t_1^2}\right)\right)

\]

At the point \( (h, k) \) where all such chords are concurrent, substituting \( x = h \) and \( y = k \):

\[

k \left(t_1^2 + 1\right) = -t_1 \left(h - 2 \left(t_1^2 + \frac{1}{t_1^2}\right)\right)

\]

Since this must be true for all \( t_1 \), the coefficients of \( t_1 \) and the constant terms must match:

\[

h = -2 \quad \text{and} \quad k = -4

\]

### Conclusion

The point \( P \) is \( (-2, -4) \). Thus, the sum of the abscissa and ordinate of \( P \) is:

\[

-2 + (-4) = -6

\]

Therefore, the sum of the abscissa and ordinate of the point \( P \) is \(\boxed{-6}\).