Question

Value of ' 𝑡 ' for which intercept cut off by the chord 𝑥 + 2𝑦 + 𝑡 = 0 of the parabola � �2 =6𝑦 subtends a right angle at its vertex, is

• A. 𝑡 = −6
• B. t = −12
• C. t = 12
• D. 𝑡 ∈ 𝜙

Answer 1

Answer: (B)

t = -12

Answer 2

The equation of the parabola is $x^2 = 6y$, with vertex $V(0,0)$. The chord is given by $x + 2y + t = 0$. To find the intersection points, substitute $y = x^2/6$ into the chord equation: $x + 2(x^2/6) + t = 0$ $x + x^2/3 + t = 0$ $x^2 + 3x + 3t = 0$

Let the intersection points be $P(x_1, y_1)$ and $Q(x_2, y_2)$. From Vieta's formulas: $x_1 + x_2 = -3$ $x_1 x_2 = 3t$

The corresponding y-coordinates are $y_1 = x_1^2/6$ and $y_2 = x_2^2/6$. For the chord to subtend a right angle at the vertex, the vectors $\vec{VP}$ and $\vec{VQ}$ must be perpendicular. Their dot product must be zero: $\vec{VP} \cdot \vec{VQ} = x_1 x_2 + y_1 y_2 = 0$ Substituting $y_i = x_i^2/6$: $x_1 x_2 + (x_1^2/6)(x_2^2/6) = 0$ $x_1 x_2 + \frac{(x_1 x_2)^2}{36} = 0$

Substitute $x_1 x_2 = 3t$: $3t + \frac{(3t)^2}{36} = 0$ $3t + \frac{9t^2}{36} = 0$ $3t + \frac{t^2}{4} = 0$ $t(3 + t/4) = 0$

This gives two possible values for $t$: $t=0$ or $3 + t/4 = 0 \implies t = -12$.

For distinct intersection points, the discriminant of $x^2 + 3x + 3t = 0$ must be positive: $D = 3^2 - 4(1)(3t) = 9 - 12t > 0 \implies 12t < 9 \implies t < 3/4$. Both $t=0$ and $t=-12$ satisfy this.

If $t=0$, the equation is $x^2 + 3x = 0$, giving $x=0$ or $x=-3$. The intersection points are $(0,0)$ (the vertex) and $(-3, 3/2)$. A chord connecting the vertex to another point does not typically form a subtended angle in the standard sense.

If $t=-12$, the equation is $x^2 + 3x - 36 = 0$. The discriminant is $9 - 4(1)(-36) = 9 + 144 = 153 > 0$, ensuring two distinct intersection points, neither of which is the vertex. This case satisfies the condition for a right angle subtended by a chord at the vertex. Therefore, the value of $t$ is -12.