Solving for Chord Property on Parabola
We are given a parabola defined by the equation $y^2 = 12x$ and two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on it. The chord connecting these points, $P_1P_2$, subtends a right angle at the vertex of the parabola, $V(0,0)$. We need to find the value of the expression $x_1x_2 - y_1y_2$.
Parabola Analysis
The standard equation of a parabola with vertex at the origin is $y^2 = 4ax$. Comparing this with the given equation $y^2 = 12x$, we find $4a = 12$, which implies $a = 3$. The vertex $V$ is at $(0,0)$.
Using Parametric Coordinates
Points on the parabola $y^2 = 4ax$ can be represented parametrically as $(at^2, 2at)$. For our parabola $y^2 = 12x$ (where $a=3$), the points are of the form $(3t^2, 6t)$.
- Let $P_1$ correspond to parameter $t_1$, so $P_1 = (3t_1^2, 6t_1)$. This means $x_1 = 3t_1^2$ and $y_1 = 6t_1$.
- Let $P_2$ correspond to parameter $t_2$, so $P_2 = (3t_2^2, 6t_2)$. This means $x_2 = 3t_2^2$ and $y_2 = 6t_2$.
Condition for Right Angle at Vertex
The chord $P_1P_2$ subtending a right angle at the vertex $V(0,0)$ means the line segments $VP_1$ and $VP_2$ are perpendicular.
The slope of $VP_1$ is $m_1 = \frac{y_1 - 0}{x_1 - 0} = \frac{6t_1}{3t_1^2} = \frac{2}{t_1}$ (assuming $t_1 \neq 0$).
The slope of $VP_2$ is $m_2 = \frac{y_2 - 0}{x_2 - 0} = \frac{6t_2}{3t_2^2} = \frac{2}{t_2}$ (assuming $t_2 \neq 0$).
The condition for perpendicularity is $m_1 m_2 = -1$. $ \implies \left(\frac{2}{t_1}\right) \left(\frac{2}{t_2}\right) = -1 $ $ \implies \frac{4}{t_1 t_2} = -1 $ $ \implies t_1 t_2 = -4 $
Calculating the Expression
We need to evaluate $x_1x_2 - y_1y_2$. Using the parametric forms:
- $x_1x_2 = (3t_1^2)(3t_2^2) = 9 (t_1 t_2)^2$
- $y_1y_2 = (6t_1)(6t_2) = 36 (t_1 t_2)$
Substitute the derived value $t_1 t_2 = -4$:
- $x_1x_2 = 9 (-4)^2 = 9(16) = 144$
- $y_1y_2 = 36 (-4) = -144$
Now, compute the final expression:
$ x_1x_2 - y_1y_2 = 144 - (-144) = 144 + 144 = 288 $Similar Questions
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- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
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Important Questions from Coordinate Geometry
- Let a line L passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1}$ and $\frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0}$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point $S(1, 0, -1)$ intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
