Let A and B be two distinct point on the parabola y2 = 4x. If the axis of the parabola touches the circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

Engineering

Mathematics

Standard Parabola

Question

Let A and B be two distinct point on the parabola y² = 4x. If the axis of the parabola touches the circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

$\frac{1}{r}$

$\frac{2}{r}$

$\frac{- 2}{r}$

$\frac{- 1}{r}$

y-coordinates of the centre = radius = r

$∴$ t₁ + t₂ = r or – 2

$∴$ slopes of AB = $\frac{2}{t_{1} + t_{2}}$

$= \frac{2}{r} or \frac{- 2}{r}$

Problem Explanation

Given a parabola $y^{2} = 4 x$ , let A and B be two distinct points on it. The circle with AB as diameter has radius r, and it touches the axis of the parabola (which is the x-axis). We need to find the possible slopes of line AB.

Step 1: Parameterize Points on the Parabola

Any point on the parabola $y^{2} = 4 x$ can be written in parametric form as $(t_{1}^{2}, 2 t_{1})$ and $(t_{2}^{2}, 2 t_{2})$ for parameters $t_{1}$ and $t_{2}$ . Let these be points A and B respectively.

Step 2: Find the Slope of AB

The slope m of line joining A and B is:

$m = \frac{2 t_{2} - 2 t_{1}}{t_{2}^{2} - t_{1}^{2}} = \frac{2 (t_{2} - t_{1})}{(t_{2} - t_{1}) (t_{2} + t_{1})} = \frac{2}{t_{1} + t_{2}}$

So, $m = \frac{2}{t_{1} + t_{2}}$ . Let $h = t_{1} + t_{2}$ , then $m = \frac{2}{h}$ .

Step 3: Equation of the Circle with AB as Diameter

The center of the circle is the midpoint of AB. Coordinates of center C are:

$C = (\frac{t_{1}^{2} + t_{2}^{2}}{2}, \frac{2 t_{1} + 2 t_{2}}{2}) = (\frac{t_{1}^{2} + t_{2}^{2}}{2}, t_{1} + t_{2}) = (\frac{t_{1}^{2} + t_{2}^{2}}{2}, h)$

The radius r of the circle is half the length of AB. However, we are given that the circle has radius r and touches the x-axis.

Step 4: Condition for the Circle to Touch the x-axis

The x-axis is a horizontal line (y=0). For a circle with center (x₀, y₀) to touch the x-axis, the perpendicular distance from the center to the x-axis must equal the radius. Since the x-axis is y=0, this distance is |y₀|. Therefore:

$| y_{0} | = r$

Here, the y-coordinate of the center is h. So, |h| = r.

Thus, $h = r$ or $h = - r$ .

Step 5: Find the Slope m

From Step 2, we have $m = \frac{2}{h}$ .

Case 1: If h = r, then $m = \frac{2}{r}$ .

Case 2: If h = -r, then $m = \frac{2}{- r} = - \frac{2}{r}$ .

Final Answer

The slope of the line joining A and B can be $\frac{2}{r}$ or $- \frac{2}{r}$ .

Therefore, the correct options are: $\frac{2}{r}$ and $\frac{- 2}{r}$ .

Important Formulae

Parametric form of parabola $y^{2} = 4 a x$ : $(a t^{2}, 2 a t)$ . Here a=1.
Slope m of line joining (x₁, y₁) and (x₂, y₂): $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
Center of circle with diameter endpoints (x₁, y₁) and (x₂, y₂): $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
Distance from point (x₀, y₀) to line y=0 is |y₀|.

Detailed Solution

Problem Explanation

Step 1: Parameterize Points on the Parabola

Step 2: Find the Slope of AB

Step 3: Equation of the Circle with AB as Diameter

Step 4: Condition for the Circle to Touch the x-axis

Step 5: Find the Slope m

Final Answer

Related Topics

Important Formulae

Detailed Solution

Problem Explanation

Step 1: Parameterize Points on the Parabola

Step 2: Find the Slope of AB

Step 3: Equation of the Circle with AB as Diameter

Step 4: Condition for the Circle to Touch the x-axis

Step 5: Find the Slope m

Final Answer

Related Topics

Important Formulae

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