find the points of intersection of the line x-y+2=0 and the circle 3Xsquare + 3Ysquare -29x -19y +56 =0 - Maths - Conic Sections

Question

find the points of intersection of the line x-y+2=0 and the
circle 3Xsquare + 3Ysquare -29x -19y +56 =0 also find the length of
the chord intersepted

Lalit Mehra · Accepted Answer

Dear Student!

Equation of the given line is x – y + 2 = 0 ...(1)

Equation of the circle is 3x² + 3y² – 29x – 19y + 56 = 0 ...(2)

Solving (1) and (2), we get

$3 x^{2} + 3 (x + 2)^{2} - 29 x - 19 (x + 2) + 56 = 0 \Rightarrow 3 x^{2} + 3 (x^{2} + 4 x + 4) - 29 x - 19 x - 38 + 56 = 0 \Rightarrow 3 x^{2} + 3 x^{2} + 12 x + 12 - 29 x - 19 x - 38 + 56 = 0 \Rightarrow 6 x^{2} - 36 x + 30 = 0 \Rightarrow x^{2} - 6 x + 5 = 0 \Rightarrow x^{2} - 5 x - x + 5 = 0 \Rightarrow x (x - 5) - 1 (x - 5) = 0 \Rightarrow (x - 5) (x - 1) = 0 \Rightarrow x - 5 = 0 or x - 1 = 0 \Rightarrow x = 5 or x = 1$

When x = 5, we get

5 – y + 2 = 0 (Using (1))

∴ y = 7

When x = 1, we get

1 – y + 2 = 0 (Using (1))

∴ y = 3

Coordinates of the end points of the chord intercepted by the given line and the given circle are (5, 7) and (1, 3).

∴ Length of the chord $= \sqrt{(5 - 1)^{2} + (7 - 3)^{2}} = \sqrt{(4)^{2} + (4)^{2}} = \sqrt{32} = 4 \sqrt{2}$

Cheers!

80

find the points of intersection of the line x-y+2=0 and the circle 3Xsquare + 3Ysquare -29x -19y +56 =0. also find the length of the chord intersepted.