Question

The straight line ax + by = 1 makes with the curve px2 + 2axy + qy2 = r, a chord which subtends a right angle at the origin. Then (1) r(b2 + q2) = p +a (2) r(62 + p2) = p + q. (3) ra? + b2) = p +9 (4) (a2 + p)r = q + b​

Answer 1

Correct answer is p + q = r (a² + b²)

A chord subtends a right angle at the origin. So,

px² + 2 axy + qy² = r . And

ax + by = 1 (given)

So,

px²+ 2 axy + qy² - r ($\frac{ax + by}{1}$)² = 0

By further solving we get-

x² (p-a²r) + y² (q-b²r) + xy (2a-2abr) = 0

A chord makes a 90° angle so lines are perpendicular.

So sum of the coefficient of x² + y² is equals to zero.

Coefficient of x² = p-a²r and

Coefficient of y² = q-b²r

p + q = r (a² + b²)

Answer 2

Correct answer is p + q = r (a² + b²)

A chord subtends a right angle at the origin. So,

px² + 2 axy + qy² = r . And

ax + by = 1 (given)

So,

px²+ 2 axy + qy² - r (\frac{ax + by}{1}

1

ax+by

)² = 0

By further solving we get-

x² (p-a²r) + y² (q-b²r) + xy (2a-2abr) = 0

A chord makes a 90° angle so lines are perpendicular.

So sum of the coefficient of x² + y² is equals to zero.

Coefficient of x² = p-a²r and

Coefficient of y² = q-b²r

p + q = r (a² + b²)