The feet of the perpendiculars from a point \(P_1\) to the sides of a triangle \(ABC\) lie on a straight line \(\lambda_1\); prove that \(P_1\) lies on the circle circumscribing the triangle \(ABC\). Prove that, if \(P_2, \lambda_2\) and \(P_3, \lambda_3\) are related in the same way, the triangle \(P_1P_2P_3\) is similar to the triangle formed by \(\lambda_1, \lambda_2, \lambda_3\).
Prove that the operations of inversion with respect to two coplanar circles in succession are commutative if the circles cut one another orthogonally. \(A', B', C'\) are the inverses of three fixed points \(A, B, C\) with regard to a variable circle whose centre is \(P\). Find the locus of \(P\) when the triangle \(A'B'C'\) is right angled.
Given a focus and the corresponding directrix of a conic and also the eccentricity, obtain a geometrical construction for the two points in which the conic is cut by any straight line.
A variable line moves in a plane so that the intercepts made on it by the sides of a fixed coplanar triangle bear constant ratios to one another. Show that the line envelopes a parabola inscribed in the fixed triangle.
A point moves on a given plane so that the line joining it to a fixed point not in the plane makes a given angle with a fixed line in the plane. Show that the locus of the point is a hyperbola and that as the given angle varies the corresponding hyperbolas are coaxal.
The points in a plane are displaced so that the point \((x,y)\) referred to rectangular coordinates takes the position \((X,Y)\), where \(X=px+qy, Y=rx+sy\). Show that a unit square in any position becomes a parallelogram of area \(ps \sim qr\), and that the parallelogram has the sum of the squares of the lengths of its sides constant. What is the least possible angle between the sides of the parallelogram?
The three sides of a varying triangle touch the parabola \(y^2=4ax\), and two of the vertices lie on the confocal parabola \(y^2=4(a+\lambda)(x+\lambda)\); prove that the third vertex lies on the confocal \(y^2=4(a+\mu)(x+\mu)\), where \(a\mu=4\lambda(a+\lambda)\).
Show that the focal radius vector \(r\) of a point on an ellipse, the angle \(\theta\) made by the vector with the major axis and the eccentric angle \(\phi\) of the point are connected by the relations \[ \frac{l}{r} = 1+e\cos\theta, \quad r=a(1-e\cos\phi), \quad \tan\frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{\phi}{2}. \] Show also that the maximum value of \(\theta-\phi\) is \(2\sin^{-1}\sqrt{\frac{a-b}{a+b}}\), and that it occurs when \(\theta+\phi=\pi, r=b\).
Find the equation to the pair of tangents drawn from a point to the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-1=0\). A pair of tangents to any confocal of \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-1=0\) pass respectively through the fixed points \((0,c_1)\) and \((0,c_2)\): show that the intersection of the tangents lies on the circle \[ (x^2+y^2-a^2+b^2)(c_1+c_2) = 2y(c_1c_2-a^2+b^2). \]
Find the condition that the line \(lx+my+n=0\) touches the conic \[ ax^2+by^2+c+2fy+2gx+2hxy=0. \] Show that, if \(x\cos\psi+y\sin\psi=p_1\), \(x\cos\psi+y\sin\psi=p_2\) represent a variable pair of parallel tangents of a fixed conic, the lines \[ x\cos\psi+y\sin\psi=\lambda p_1 + \mu p_2, \quad x\cos\psi+y\sin\psi=\lambda p_2 + \mu p_1, \] where \(\lambda, \mu\) are constants, envelope another fixed conic with parallel asymptotes.