Area of Triangle with Coordinates of Vertices Given

Q.

The coordinates of the incentre of the triangle having sides $3x-4y=0,5x+12y=0$ and $y-15=0$ are

1. $(-1,8)$

2. $(1,–8)$

3. $(2,6)$

4. None of these

Q. The value of k for which the points (2, -3), (k, -1) and (0, 4) are collinear, is .

1. $k=-\frac{10}{7}$
2. $k=\frac{10}{7}$
3. $k=\frac{18}{7}$
4. $k=-\frac{18}{7}$

Q. Tangent at a point $P_1$ (other than origin) on the curve $y=x^3$ meets the curve again at $P_2.$ The tangent at $P_2$ meets the curve at $P_3$ and so on. Then which of the following is/are correct?

1. Abscissa of $P_1,P_2,\cdots ,P_3$ are in G.P.
   
2. Ordinate of $P_1,P_2,\cdots ,P_3$ are in G.P.
3. Ratio of area of $\Delta P_1{P}_2{P}_3$ and $\Delta P_2{P}_3{P}_4$ is $\frac{1}{8}$
4. Ratio of area of $\Delta P_1{P}_2{P}_3$ and $\Delta P_2{P}_3{P}_4$ is $\frac{1}{16}$

Q. The area of a triangle with vertices at (−4, −1), (1, 2) and (4, −3) is
(a) 17
(b) 16
(c) 15
(d) none of these

Q. Verify that the area of the triangle with vertices (2, 3), (5, 7) and (− 3 − 1) remains invariant under the translation of axes when the origin is shifted to the point (−1, 3).

Q. If the points (a, 0), (at₁², 2at₁) and (at₂², 2at₂) are collinear, write the value of t₁ t₂.

Q. Verify that the area of the triangle with vertices (4, 6), (7, 10) and (1, −2) remains invariant under the translation of axes when the origin is shifted to the point (−2, 1).