Let one end of a focal chord of the parabola y² = 16x be (16, 16). If P(α, β) divides this focal cho

Q: Why does $t_1 \cdot t_2 = -1$? Can't I just use the slope formula?

You absolutely can use the slope formula! The slope of the line joining $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$ is $\frac{2a(t_2 - t_1)}{a(t_2^2 - t_1^2)} = \frac{2}{t_1 + t_2}$. Since the line passes through the focus $(a, 0)$, the slope is also $\frac{2at_1 - 0}{at_1^2 - a} = \frac{2t_1}{t_1^2 - 1}$. Equating these leads to $t_1 t_2 = -1$. It is a powerful shortcut that saves precious minutes in the exam hall.

Q: Wait, why do we have to check two cases? Doesn't 'divides internally' imply a specific order?

In coordinate geometry, unless the problem explicitly states 'from $A$ to $B$', the segment is just a set of points. The ratio $5:2$ defines the position relative to the endpoints. Since the segment is undirected, $P$ could be closer to $A$ or closer to $B$. Always check both to be safe!

Q: Is there a faster way to find $\alpha + \beta$ without calculating individual coordinates?

Yes! You can use the property of the centroid or weighted averages. Since $P$ divides $AB$ in ratio $m:n$, the coordinates are $\frac{mB + nA}{m+n}$. The sum $\alpha + \beta$ is just the sum of the coordinates of the weighted average. It is elegant, but calculating the coordinates directly is often less prone to silly arithmetic errors.

Question

Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$. If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5 : 2$, then the minimum value of $\alpha + \beta$ is equal to :

Options:

(A) 16

(B) 5

(C) 7

(D) 22

Sigma Percentile · Accepted Answer

### Analyzing the Parabola Geometry

We begin with the given parabola equation:
$$y^2 = 16x$$

By comparing this to the standard form $y^2 = 4ax$, we identify $4a = 16$, which yields $a = 4$. Consequently, the focus of the parabola is located at $S(4, 0)$.

### Identifying the Focal Chord

Any point on this parabola can be represented by the parametric coordinates $(at^2, 2at)$. Given $a = 4$, the coordinates are $(4t^2, 8t)$. For point $A(16, 16)$, we equate the $y$-coordinates:
$$8t_1 = 16 \implies t_1 = 2$$

For any focal chord with endpoints defined by parameters $t_1$ and $t_2$, the fundamental property is:
$$t_1 \cdot t_2 = -1$$

Substituting $t_1 = 2$, we find the parameter for the other endpoint $B$:
$$t_2 = -\frac{1}{2}$$

### Calculating Coordinates of Point B

Using the parameter $t_2 = -\frac{1}{2}$, we calculate the coordinates of point $B$:
$$x_B = 4\left(-\frac{1}{2}\right)^2 = 1$$
$$y_B = 8\left(-\frac{1}{2}\right) = -4$$

Thus, the coordinates of point $B$ are $(1, -4)$.

### Applying the Section Formula

We seek a point $P(\alpha, \beta)$ that divides the chord $AB$ in a ratio of $5:2$. We must consider both possible directions of division.

**Case 1: $P$ divides $AB$ in ratio $5:2$**
Using the section formula:
$$\alpha = \frac{5(1) + 2(16)}{5+2} = \frac{37}{7}$$
$$\beta = \frac{5(-4) + 2(16)}{5+2} = \frac{12}{7}$$
The sum is $\alpha + \beta = \frac{37+12}{7} = \frac{49}{7} = 7$.

**Case 2: $P$ divides $BA$ in ratio $5:2$**
Using the section formula:
$$\alpha = \frac{5(16) + 2(1)}{5+2} = \frac{82}{7}$$
$$\beta = \frac{5(16) + 2(-4)}{5+2} = \frac{72}{7}$$
The sum is $\alpha + \beta = \frac{82+72}{7} = \frac{154}{7} = 22$.

### Final Conclusion

Comparing the two possible sums, $7$ and $22$, the problem asks for the minimum value. Therefore, the **minimum value is 7**.

Animated Solution for Mathematics - Conic Sections: Let one end of a focal chord of the parabola $y^{2} = 16 x$ be $(16, 16)$ . If $P (α, β)$ divides this focal chord internally in the ratio $5 : 2$ , then the minimum value of $α + β$ is equal to :

Select Answer:

Visualized Solution

Identify the Parabola & Focus

Parametric Form of Point $A$

Finding Parameter $t_{1}$

The Focal Chord Property

Finding Parameter $t_{2}$

Setup for Point $B$

Evaluating Point $B$

The Internal Division Trap

Case 1: Ratio from $A$ to $B$

Case 1 Calculation

Case 2: Ratio from $B$ to $A$

Case 2 Calculation

Final Conclusion

The Sigma Insight: Parametric Equations

Analyzing the Parabola Geometry

Identifying the Focal Chord

Calculating Coordinates of Point B

Applying the Section Formula

Final Conclusion

Similar Questions

Let $O$ be the vertex and $Q$ be any point on the parabola, $x^{2} = 8 y$ . If the point $P$ divides the line segment $O Q$ internally in the ratio $1 : 3$ , then locus of $P$ is:

A line passing through the point $P (5, 5)$ intersects the ellipse $\frac{x ^{2}}{36} + \frac{y ^{2}}{25} = 1$ at $A$ and $B$ such that $(P A) \cdot (P B)$ is maximum. Then $5 (P A^{2} + P B^{2})$ is equal to :

Select Answer:

Visualized Solution

Identify the Parabola & Focus

Parametric Form of Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Finding Parameter .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t1​

The Focal Chord Property

Finding Parameter .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t2​

Setup for Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

Evaluating Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

The Internal Division Trap

Case 1: Ratio from .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

Case 1 Calculation

Case 2: Ratio from .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Case 2 Calculation

Final Conclusion

The Sigma Insight: Parametric Equations

Analyzing the Parabola Geometry

Identifying the Focal Chord

Calculating Coordinates of Point B

Applying the Section Formula

Final Conclusion

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let O be the vertex and Q be any point on the parabola, x2=8y. If the point P divides the line segment OQ internally in the ratio 1:3, then locus of P is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A line passing through the point P(5​,5​) intersects the ellipse 36x2​+25y2​=1 at A and B such that (PA)⋅(PB) is maximum. Then 5(PA2+PB2) is equal to :

Parametric Form of Point $A$

Finding Parameter $t_{1}$

Finding Parameter $t_{2}$

Setup for Point $B$

Evaluating Point $B$

Case 1: Ratio from $A$ to $B$

Case 2: Ratio from $B$ to $A$

Let $O$ be the vertex and $Q$ be any point on the parabola, $x^{2} = 8 y$ . If the point $P$ divides the line segment $O Q$ internally in the ratio $1 : 3$ , then locus of $P$ is:

A line passing through the point $P (5, 5)$ intersects the ellipse $\frac{x ^{2}}{36} + \frac{y ^{2}}{25} = 1$ at $A$ and $B$ such that $(P A) \cdot (P B)$ is maximum. Then $5 (P A^{2} + P B^{2})$ is equal to :