Question

31. Abscissa of two point P and Q on parabola y2 = 8x are roots of the equation x2 - 17x+11=0,
let tangents at P and Q meet at point T, then the distance of T from focus of parabola is
(A) 7
(B) 6
(C) 5
(D)4​

Answer 1

Given:

1. Equation of parabola y² = 8x
2. Quadratic equation x² - 17x+11=0
3. Tangents at P and Q meet at point T

To find:

The distance of T from focus of parabola

Answer:

• The given quadratic equation is x² - 17x+11=0.

• Roots of the above quadratic are x1 = 0.674 and x2 = 16.32.
• Considering the equation of parabola, at  x1 = 0.674, we get y1 = 2.32 and at x2 = 16.32, we get y2= 11.43.
• The points are (0.674 ,2.32) and (16.32, 11.43).
• The parabola y² = 8x is of the form y² = 4ax, where a =2.
• The equation of tangent to the parabola is yy1 = 2a (x1 + x).
• The two tangent equations are thus -
• 2.32y = 4(0.674 +x).    -(1)
• 11.43y = 4(16.32+x)     -(2)
• The point of intersection of the two tangents is (3.308, 6.86).
• The focus of the parabola is given by (a, 0) = (2, 0)
• Distance between (3.308, 6.86) and (2, 0) is -
• $d= \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$
• d = 6.98 ≈ 7

Thus the distance of T from focus of parabola is (A) 7.