$$\begin{gathered} {\mathrm{x}}^2 + {\mathrm{y}}^2 - 8\mathrm{x} + 40 = 0 \\ 5{\mathrm{x}}^2 + 5{\mathrm{y}}^2 -25\mathrm{x} + 80 = 0 \\ {\mathrm{x}}^2 + {\mathrm{y}}^2 - 8\mathrm{x} + 16\mathrm{y} + 160 = 0 \\ \mathrm{From} \mathrm{the} \mathrm{point} \mathrm{P} \mathrm{are} \mathrm{equal}, \mathrm{then} \mathrm{P} = ? \end{gathered}$$

• $\left(8, \frac{15}{2}\right)$

• $\left(- 8, \frac{15}{2}\right)$

• $\left(8, \frac{- 15}{2}\right)$

• $\left(- 8, \frac{- 15}{2}\right)$

The equation of the circle concentric with the circle x² + y² - 6x + 12y + 15 = 0 and of double its area is

• x² + y² - 6x +12y - 15 = 0

• x² + y² - 6x +12y - 30 = 0

• x² + y² - 6x +12y - 25 = 0

• x² + y² - 6x +12y - 20 = 0

If the circle x² + y² + 2x + 3y + 1 = 0 cuts another circle x² + y² + 4x + 3y + 2 = 0 in A and B, then the equation of the circle with AB as a diameter is

• x² + y²  + x + 3y + 1 = 0

• 2x² + 2y²  + 2x + 6y + 1 = 0

• x² + y²  + x + 6y + 1 = 0

• 2x² + 2y²  + x + 3y + 1 = 0

The equation of the hyperbola which passes through the point (2, 3) and has the asymptotes 4x + 3y - 7 = 0 and x - 2y - 1 = 0 is

• 4x² + 5xy - 6y² - 11x + 11y + 50 = 0

• 4x² + 5xy - 6y² - 11x + 11y - 43 = 0

• 4x² - 5xy - 6y² - 11x + 11y + 57 = 0

• x² - 5xy - y² - 11x + 11y - 43 = 0

The product of the perpendicular distances from any point on the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ to its asymtotes is

• $\frac{{\mathrm{a}}^2{\mathrm{b}}^2}{{\mathrm{a}}^2 - {\mathrm{b}}^2}$

• $\frac{{\mathrm{a}}^2{\mathrm{b}}^2}{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\frac{{\mathrm{a}}^2 + {\mathrm{b}}^2}{{\mathrm{a}}^2{\mathrm{b}}^2}$

• $\frac{{\mathrm{a}}^2 - {\mathrm{b}}^2}{{\mathrm{a}}^2{\mathrm{b}}^2}$

If the lines 2x + 3y +12 = 0, x - yy + k = 0 are conjugate with respect to the parabola y² = 8x, then k is equal to

• 10

• $\frac{7}{2}$

• - 12

• - 2

Find the equation to the parabola, whose axis parallel to they-axis and which passes through the points (0, 4), (1, 9) and (4, 5) is

• y = - x² + x + 4

• y = - x² + x + 1

• $\mathrm{y} = \frac{- 19}{12}{\mathrm{x}}^2 + \frac{79}{12}\mathrm{x} + 4$

• $\mathrm{y} = \frac{- 19}{12}{\mathrm{x}}^2 + \frac{89}{12}\mathrm{x} + 4$

18.

If ∝, ß, y are the roots of the equation x³ - 6x² + 11x - 6 = 0 and if a = ∝² + ß² + γ², b = ∝ß + ßγ + γ∝ and  c = (∝ + ß)(ß + γ)(γ + ∝), then the correct inequality among the following is

• $\mathrm{a} < \mathrm{b} < \mathrm{c}$

• $\mathrm{b} < \mathrm{a} < \mathrm{c}$

• $\mathrm{b} < \mathrm{c} < \mathrm{a}$

• $\mathrm{c} < \mathrm{a} < \mathrm{b}$

A plane meets the coordinate axes at A, B, C so that the centroid of the triangle ABC is (1, 2, 4). Then, the equation of the plane is

• x + 2y +4z =12

• 4x + 2y + z = 12

• x + 2y + 4z = 3

• 4x + 2y + z = 3

If (2, 3, - 3) is one end of a diameter of the sphere x² + y² + z² - 6x - 12y - 2z + 20 = 0, then the other end of the diameter is

• (4, 9, - 1)

• (4, 9, 5)

• (- 8, - 15, 1)

• (8, 15, 5)