do it with full step by step calculations, straight line L:2x+y+5=0 cuts the circle C:x2 +y2-4x+3y-10= 0 at two points P and Q. Straight line Lz: 3x + 2y-16= 0 cuts the cirele C at two points R and S. (a) Find the coordinates of P, q, R and S. (b) Find the perimeter of the quadrilateral PqRS

(a) To find the coordinates of P, Q, R, and S, we need to solve the equations of the lines and the circle simultaneously.

For points P and Q:

The equation of line L is 2x + y + 5 = 0. We can rearrange this to y = -2x - 5.

Substitute y = -2x - 5 into the equation of the circle C: x^2 + y^2 - 4x + 3y - 10 = 0.

This gives: x^2 + (-2x - 5)^2 - 4x + 3(-2x - 5) - 10 = 0.

Solving this quadratic equation will give the x-coordinates of points P and Q. The corresponding y-coordinates can be found by substituting the x-coordinates into the equation y = -2x - 5.

Similarly, for points R and S:

The equation of line Lz is 3x + 2y - 16 = 0. We can rearrange this to y = (16 - 3x) / 2.

Substitute y = (16 - 3x) / 2 into the equation of the circle C: x^2 + y^2 - 4x + 3y - 10 = 0.

This gives: x^2 + ((16 - 3x) / 2)^2 - 4x + 3((16 - 3x) / 2) - 10 = 0.

Solving this quadratic equation will give the x-coordinates of points R and S. The corresponding y-coordinates can be found by substituting the x-coordinates into the equation y = (16 - 3x) / 2.

(b) To find the perimeter of the quadrilateral PQRS, we need to find the distances between the points P, Q, R, and S.

The distance between two points (x1, y1) and (x2, y2) is given by the formula √((x2 - x1)^2 + (y2 - y1)^2).

Calculate the distances PQ, QR, RS, and SP, and add them together to find the perimeter of the quadrilateral PQRS.