Math Problems Vault

Problem 12.25

Find the position vector of the image of an arbitrary point $M(\mathbf{r})$ under the given transformation of the plane:

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(1)

a homothety with center at the point $M_{0}\left(\mathbf{r}_{0}\right)$ and ratio $k \neq 0$

(2)

a point reflection (central symmetry) with respect to the point $M_{0}\left(\mathbf{r}_{0}\right)$

(3)

a translation by the vector $\mathbf{a}$

(4)

an orthogonal projection onto the line $\mathbf{r} = \mathbf{r}_{0} + t\mathbf{a}$

(5)

a reflection with respect to the line $\mathbf{r} = \mathbf{r}_{0} + t\mathbf{a}$

(6)

a contraction toward the line $\mathbf{r} = \mathbf{r}_{0} + t\mathbf{a}$ with ratio $\lambda > 0$ .

Problem 12.26

An affine transformation takes three points $A$ , $B, C$ , not lying on one line, respectively to the points $B$ , $C, A$ . Find the fixed points of this transformation. Under what necessary and sufficient condition is the transformation orthogonal?

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Problem 12.27

An affine transformation takes the vertices of a triangle $ABC$ to the midpoints $K, L, M$ of the sides opposite them. Find the images of the points $K, L, M$ and of the point $O$ of intersection of the medians of the triangle $ABC$ . Determine the geometric meaning of this transformation.

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Problem 12.28

(1)(2)(3)

Prove that:

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(1)

if $A$ and $B$ are two distinct fixed points of an affine transformation, then all points of the line $AB$ are fixed as well

(2)

if an affine transformation $f$ has a unique fixed point, then all invariant lines (if they exist) pass through this point

(3)

the point of intersection of two invariant lines of an affine transformation is fixed.

Problem 12.29

Prove that an affine transformation having a pencil of invariant lines intersecting at a point $M$ is a homothety with center at the point $M$ .

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Problem 12.30

Prove that a linear transformation of the plane is affine if and only if the image of every nonzero vector is nonzero.

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Problem 12.31

Prove that two tangent lines to an ellipse (or a hyperbola) are parallel if and only if the points of tangency and the center of the curve lie on one line.

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Problem 12.32

Prove that if an ellipse is tangent to a side of a parallelogram circumscribed about it at its midpoint, then it is tangent to the remaining sides of this parallelogram also at their midpoints.

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Problem 12.33

A quadrilateral $ABCD$ is circumscribed about an ellipse with center $O$ . Prove that the sum of the areas of the triangles $OAB$ and $OCD$ equals the sum of the areas of the triangles $OBC$ and $OAD$ .

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Problem 12.34

Prove that the vertices of a rhombus circumscribed about an ellipse lie on the axes of symmetry of this ellipse.

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Problem 12.35

Points $A, B, C, D$ lie on an ellipse with center $O$ , and the areas of the sectors $AOB$ and $COD$ are equal. Prove that the areas of the triangles $AOB$ and $COD$ are also equal.

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Problem 12.36

Points $A$ and $B$ lie on an ellipse with center $O$ , whose major and minor semi-axes have lengths $a$ and $b$ respectively. Find the area of the sector $AOB$ , if the angle $AOB$ equals $\varphi$ , $0 < \varphi \leqslant \pi$ , and the points $A$ and $B$ are symmetric with respect to the major axis of the ellipse.

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