The number of points on the line 3x+4y=5 which are at a distance of sec^2(a)+2cosec^2(a) where a belongs to real numbers, from the point (1,3)

To solve the problem of finding the number of points on the line represented by the equation \(3x + 4y = 5\) that are at a specific distance from the point (1, 3), where that distance is given by \( \sec^2(a) + 2\csc^2(a) \), we need to break it down into manageable parts.

Understanding the Line Equation

The equation \(3x + 4y = 5\) can be rearranged into slope-intercept form, which is easier to work with for visualization. By solving for \(y\), we get:

\(4y = -3x + 5\)

\(y = -\frac{3}{4}x + \frac{5}{4}\)

This indicates that the line has a slope of \(-\frac{3}{4}\) and a y-intercept of \(\frac{5}{4}\).

Calculating the Distance

The distance we are interested in is \(d = \sec^2(a) + 2\csc^2(a)\). To understand this expression, we can recall the definitions of secant and cosecant:

• Secant: \(\sec(a) = \frac{1}{\cos(a)}\)
• Cosecant: \(\csc(a) = \frac{1}{\sin(a)}\)

Thus, we can express the distance as:

\(d = \frac{1}{\cos^2(a)} + 2 \cdot \frac{1}{\sin^2(a)}\)

Finding Points on the Line at a Given Distance

Next, we need to determine points \((x, y)\) on the line \(3x + 4y = 5\) that are exactly \(d\) units away from the point (1, 3). The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Applying this to our points, we have:

\(D = \sqrt{(x - 1)^2 + (y - 3)^2} = d\)

Squaring both sides, we get:

\((x - 1)^2 + (y - 3)^2 = d^2\)

Substituting the Line into the Distance Equation

Now, we substitute \(y\) from the line equation into the distance equation. Substituting \(y = -\frac{3}{4}x + \frac{5}{4}\) into the distance equation gives:

\((x - 1)^2 + \left(-\frac{3}{4}x + \frac{5}{4} - 3\right)^2 = d^2\)

Simplifying the second term:

\(-\frac{3}{4}x + \frac{5}{4} - 3 = -\frac{3}{4}x + \frac{5}{4} - \frac{12}{4} = -\frac{3}{4}x - \frac{7}{4}\)

Now we have:

\((x - 1)^2 + \left(-\frac{3}{4}x - \frac{7}{4}\right)^2 = d^2\)

Solving for Points

Expanding both terms and setting the equation equal to \(d^2\) will yield a quadratic equation in \(x\). The number of solutions to this quadratic equation will tell us how many points on the line are at the specified distance from (1, 3).

Quadratics can have 0, 1, or 2 real solutions depending on the discriminant. Thus, the final step is to analyze the discriminant of the resulting quadratic equation to determine how many points satisfy the condition.

Conclusion

In summary, by substituting the line equation into the distance formula and simplifying, we can determine the number of points on the line that are at the specified distance from the given point. The discriminant of the resulting quadratic will ultimately tell us how many such points exist. The exact value of \(d\) will depend on the specific angle \(a\) chosen, but you can explore the values of \(d\) to see how it affects the number of solutions!