Question

The equation of the straight line passing through the point of intersection of the lines represented by \(x^2 + 4xy + 3y^2 - 4x - 10y + 3 = 0\) and the point \((2,2)\) is:

• A. \(2x + 3y - 10 = 0\)
• B. \(3x + 2y - 10 = 0\)
• C. \(2x + y - 6 = 0\)
• D. \(x + 2y - 6 = 0\)

Hint:

When determining the equation of a line passing through a given point and the intersection of two curves, first solve for the intersection points of the curves and then use the point-slope form of the equation to determine the line. Always check that the line satisfies both conditions.

Answer 1

The Correct Option is B

Step 1: Start by solving the quadratic equation \(x^2 + 4xy + 3y^2 - 4x - 10y + 3 = 0\). We need to find the points of intersection of the curves represented by this quadratic equation. To do this, express it as a pair of lines and solve for the points where these lines intersect. We rewrite the quadratic equation as: \[ x^2 + 4xy + 3y^2 - 4x - 10y + 3 = 0 \] This is a second-degree equation in \(x\) and \(y\) which can be factorized to obtain the lines of intersection. 
Step 2: After factoring, we obtain the two straight lines from the quadratic equation (steps of factoring are omitted for brevity). The point of intersection of these lines will be the solution to the system. 
Step 3: Now, the point \((2, 2)\) is given to lie on the required line. We can use the point-slope form of the equation of a straight line: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is the point \((2, 2)\). To find the slope \(m\), we can substitute the known intersection points from Step 1, and use the two-point form of the equation of a line to calculate the equation. 
Step 4: After performing the calculations for the slope and substituting the point \((2, 2)\) into the equation, the equation of the straight line passing through \((2, 2)\) and the point of intersection is: \[ 3x + 2y - 10 = 0. \]