Tangents PQ and PR are drawn from the point `(alpha, beta)` to the circle `x^(2)+y^(2)=a^(2)`. Show that the area of `Delta PQR` is `(a(alpha^(2)+ beta^(2)-a^(2))^(3//2))/(alpha^(2)+ beta^(2))`.

P( asec alpha, b tan alpha ) and Q (a sec beta, b tan beta) are two given points on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2)) = 1. Show that the equation of the chord PQ is (x)/(a) cos (alpha - beta)/(2) - (y)/(b) sin (alpha + beta)/(2) = cos (alpha+beta)/(2).

If two tangents drawn from the point (alpha,beta) to the parabola y^2=4x are such that the slope of one tangent is double of the other, then prove that alpha=2/9beta^2dot

Tangents are drawn from the point (alpha, beta) to the hyperbola 3x^2- 2y^2=6 and are inclined atv angle theta and phi to the x-axis.If tan theta.tan phi=2 , prove that beta^2 = 2alpha^2 - 7 .

Find the equation of the tangent to the curve x^((2)/(3))+y^((2)/(3))=a^((2)/(3)) at the point (alpha,beta)

Prove that (cos alpha - cos beta)^(2) +(sin alpha - sin beta)^(2) = 4 "sin"^(2) (alpha-beta)/2

If (alpha +beta ) and (alpha - beta ) are the eccentric angles of the points P and Q respectively on the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 show that the equation of the chord PQ is (x)/(a) cos alpha+(y)/(b)sin alpha = cos beta .

If alpha, beta are the roots of x^(2) - px + q = 0 and alpha', beta' are the roots of x^(2) - p' x + q' = 0 , then the value of (alpha - alpha')^(2) + (beta -alpha')^(2) + (alpha - beta')^(2) + (beta - beta')^(2) is

If alpha and beta be the roots of equation x^(2) + 3x + 1 = 0 then the value of ((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2) is equal to

If alpha, beta are the roots of the equation x^(2)-px+q=0 , prove that log_(e)(1+px+qx^(2))=(alpha+beta)x-(alpha^(2)+beta^(2))/(2)x^(2)+(alpha^(3)+beta^(3))/(3)x^(3)-...

If the axis are transferred to parallel axis throught the point (alpha,beta) , then equation of the circle (x-alpha)^(2)+(y-beta)^(2)=a^(2) reduces to the form-