A point P divides the line segment joining the points A (3, -5) and B (-4, 8) such that $\frac{A P}{P B} = \frac{k}{1} .$ If P lies on the line x + y = 0, then find the value of k.

Q. Point P divides the line segment joining the point A(2, 1) and B(5,-8) such that

\frac{A P}{A B} = \frac{1}{3} .

If P lies on the line 2x-y+k=0, find the value of k.

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A rod of length l moves such that its ends A and B always lie on the lines 3x−y+5=0 and y+5=0 respectively. The locus of the point P, which divides AB internally in the ratio 2:1, is l2=1k(ax−by−5)2+9(y+5)2. Then

The correct option is A k=4, a+b=6 By internal section formula, x=2β+α3, y=−10+3α+53 ⇒α=3y+53 and β=9x−3y−56 Also, l2=AB2=(α−β)2+(3α+10)2 ⇒l2=14(3x−3y−5)2+9(y+5)2

A rod of length $l$ moves such that its ends $A$ and $B$ always lie on the lines $3 x - y + 5 = 0$ and $y + 5 = 0$ respectively. The locus of the point $P,$ which divides $A B$ internally in the ratio $2 : 1,$ is $l^{2} = \frac{1}{k} (a x - b y - 5)^{2} + 9 (y + 5)^{2} .$ Then