Question

The points $\left( x_{1},y_{1} \right)$and $\left( x_{2},y_{2} \right)\mspace{6mu}\mspace{6mu} on\mspace{6mu} y^{2} = 4x$ and the point (1, 0) are collinear. Then $x_{1}x_{2} + y_{1}y_{2} =$

• A. -1
• B. -2
• C. -3
• D. –5

Answer 1

Answer: (C)

-3

Answer 2

If t₁, t₂ are the extremities of a focal chord, then t₁.t₂ =-1

Since (1, 0) is the focus of y² = 4x.

⇒ (x₁, y₁) and (x₂,y₂) are the extremities of focal chord.

⇒ x₁x₂ + y₁y₂ = $at_{1}^{2}.at_{2}^{2} + 2at_{1}.2at_{2}$

= $a^{2}(t_{1}t_{2})^{2} + 4a^{2}(t_{1}t_{2})$

= $a^{2} - 4a^{2} = - 3a^{2} = - 3$