Parabola
Right Angle at Vertex — x₁x₂ − y₁y₂
nta_pyq_2026_jan
Grade 11
Question:
If the chord joining the points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ on the parabola $y^2=12x$ subtends a right angle at the vertex of the parabola, then $x_1x_2-y_1y_2$ is equal to
Step-by-Step Solution
Key Concept: $y^2=12x$: $a=3$. Parametric: $P_1=(3t_1^2,6t_1)$, $P_2=(3t_2^2,6t_2)$. Chord subtends right angle at $O(0,0)$: $\vec{OP_1}\cdot\vec{OP_2}=0\Rightarrow9t_1^2t_2^2+36t_1t_2=0\Rightarrow t_1t_2(t_1t_2+4)=0\Rightarrow t_1t_2=-4$.
$t_1t_2=-4$. $x_1x_2-y_1y_2=288$.
Correct Answer: 3