What is the equation of the pair of straight lines perpendicular to a given pair and passing through a point?

To find the equation of the pair of straight lines that are perpendicular to the given pair of lines represented by the equation 4xy - 2x - 6y - 3 0 and pass through the point (2, 1), we can follow these steps:

Step 1: Convert the given equation to standard form

The equation 4xy - 2x - 6y - 3 0 can be rearranged to find the slopes of the lines. This can be interpreted as a quadratic in y.

4xy - 2x - 6y - 3 0 implies 4xy - 6y - 2x - 3 0

Step 2: Find the slopes of the given lines

Using the quadratic formula, the slopes m1 and m2 of the two lines can be found by comparing it to the standard quadratic form Ay2 - By C 0.

A 4x, B 6, C 2x - 3

The slopes are given by:

m1 m2 (-B ± sqrt(B2 - 4AC)) / (2A)

Substituting the values:

m1 m2 (-6 ± sqrt(62 - 4 * 4x * (2x - 3))) / (8x)

Simplifying the discriminant:

62 - 4 * 4x * (2x - 3) 36 - 32x2 - 64x

Thus the slopes are:

m1 m2 (-6 ± sqrt(36 - 32x2 - 64x)) / (8x)

Step 3: Determine the slopes of the perpendicular lines

If the slopes of the original lines are m1 and m2, the slopes m3 and m4 of the new lines that are perpendicular to these slopes will satisfy:

m3 -1/m1 and m4 -1/m2

Step 4: Equation of the lines through point (2, 1)

Using the point-slope form, the equation of a line through the point (2,1) with slope m is:

y - 1 mx - 2

For both slopes m3 and m4 we can write the equations of the lines as:

1. For slope m3:

y - 1 m3x - 2

2. For slope m4:

y - 1 m4x - 2

Step 5: Combine the equations

The two lines can be expressed as a pair of straight lines in the following form:

y - 12 m3x - 22

y - 12 m4x - 22

Summary:

To find the exact equations, we need to substitute the values for m3 and m4 from Step 2 into the equations for the lines through (2,1). The final result will yield the equations of the two lines in the required format. Since the calculations may involve complex arithmetic depending on the value of x, you may need numerical values for x to find specific slopes. If you need a general form or specific values, please clarify!