Shift the origin to a suitable point so that the equation `y^2+4y+8x-2=0` will not contain a term in `y` and the constant term.

Shift the origin to a suitable point so that the equation 2x^(2)-4x+3y+5=0 will not contain the term in x and the constant term.

The length of the intercepts on the x and y-axes of a circle are 2a and 2b unit respectively. Prove that the locus of the centre of the circle is hyperbola whose equation x^2-y^2=a^2-b^2 . Translating the origin to a suitable point show that the equation 5x^2-4y^2-20x-8y-4=0 represents a hyperbola. Find its eccentricity, coordinates of foci and equations of the directrices.

Without rotating the original coordinate axes, to which point should origin be transferred, so that the equation x^2 + y^2-4x + 6y-7=0 is changed to an equation which contains no term of first degree?

Find the point to which the origin should be shifted after a translation of axes , so that the equation 3x^(2)+8xy+3y^(2)-2x+2y-2=0 will have no first degree terms.

Find the point to which the origin should be shifted so that the equation 3y^(2)+6y-5x-7=0 is reduced to the form y^(2)=ax .

Determine whether each pair of the following equations are solvable or not by finding the relations among the ratios of the co-efficients of the same variable and the constant terms of each pair. Also write whether the graphs of the equations of each pair are parallel or intersecting of coinciding or not: (i) 5x+3y=11, 2x-7y=-12, (ii) 6x-8y=2, 3x-4y=1, (iii) 8x-7y=0, 8x-7y=56,

Find the point to which the origin should be shifted after a translation of axis so that the following equations will have no first degree terms : (i) x^(2)+y^(2)-4x-8y+3=0 , (ii) x^(2)+y^(2)-5x+2y-5=0

Find the angle through which the axis may be turned about origin , so that the equation of curve 4x^2 + 2 sqrt3 xy +2 y^2 = 1 must not have term containing 'xy'

The slope of the normal to the parabola 3y^(2)+4y+2=x at a point it is 8. find the coordinates of the points.

If the origin is shifted to the point ((a b)/(a-b),0) without rotation, then the equation (a-b)(x^2+y^2)-2a b x=0 becomes